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... It was the first example of deterministic chaos. The Lorenz model [1] was created in 1963 owing to a series of transformations of the Navier-Stokes equations. Its solutions were interesting because of their quasi-stohastic trajectories and absence of external sources of noise. ...
... At t → ∞ all phase-space trajectories are concentrated inside a compact attractor. 1 ...
... The behavior of the system after the noise appearance demonstrates quite clearly that stochastic interference plays a significant role in describing turbulence. Lorenz [1] wanted to use his model for long-term weather forecasting. Moreover, he wanted to prove the theoretical existence of such a method. ...
Nowadays interest of the deterministic differential system of Lorentz equations is still primarily due to the problem of gas and fluid turbulence. Despite numerous existingsystems for calculating turbulent flows, new modifications of already known models are constantly being investigated. In this paper we consider the effect of stochastic additive perturbations on the Lorentz convective turbulence model. To implement this and subsequent interpretation of the results obtained, a numerical simulation of the Lorentz system perturbed by adding a stochastic differential to its right side is carried out using the programming capabilities of the MATLAB programming environment.
На сегодняшний день интерес к детерминированной дифференциальной системе уравнений Лоренца по-прежнему обусловлен прежде всего проблемой турбулентности газов и жидкости. Несмотря на большое число существующих систем для расчета турбулентных течений, постоянно исследуются новые модификации уже известных моделей. В данной работе рассматривается влияние стохастических аддитивных возмущений на модель конвективной турбулентности Лоренца. Для реализации этого и последующей интерпретации полученных результатов, осуществляется численное моделирование системы Лоренца, возмущенной за счет добавления в ее правую часть стохастического дифференциала, с использованием программных возможностей среды программирования MATLAB.
... In the form of its application most closely related to our work, the desire to generate a single-valued coarse-to-fine map restricts the choice of the number and type of coarse variables. This may be intuitively understood in the context of the Lorenz system (Lorenz, 1963); here the fine system is 3dimensional, and trajectories tend to converge on a topologically complicated, but bounded, set that can be enveloped within another bounded and contiguous region of 3-d phase space. However, it is not possible to represent the enveloping region as a graph of a single function over the space consisting of any two of the system degrees of freedom that one might wish to choose as coarse variables. ...
... The fine set of equations is given as (Lorenz, 1963) ...
... shown by Lorenz (1963). For the assumed values of σ and b , Lorenz (1963) also shows that the three fixed points become hyperbolic saddles as r crosses the linear stability threshold of 24.74 r = from below. ...
A method for model reduction in nonlinear ODE systems is demonstrated through computational examples. The method does not require an implicit separation of time-scales in the fine dynamics to be effective. From the computational standpoint, the method has the potential of serving as a subgrid modeling tool. From the physical standpoint, it provides a model for interpreting and describing history dependence in coarse-grained response of an autonomous system.
... 11 These routes are and listed in Table III B and illustrated in Figure III B. The table lists the routes in chronological order of the first applied-for license. 12 The "registered licensee" is generally a company spawned in order to separate the route from the identity of its user/operator, though some do correspond to publicly disclosed carriers. Total and route distances D tot and D geo are computed between the endpoints of the MW route, though the information channel may continue by other technologies such as fiber, free-space optical, or other RF channels for short distances beyond these endpoints, to connect to data centers. ...
... In this regard, we speculate that "dissipation" in a market viewed as a dynamical system can be identified with the noise trading provided by retail orders, and that the Lyapunov time, τ L , is of order τ L ∼ τ lat ∼ 10 ms, as measured by the exponential decay observed in our response curves over this time scale. If this view has merit, then the equity exchanges display a dramatic contrast in their Lyapunov time scale to that shown by Earth's weather, which exhibits predictability over periods measured in days [12], and to Earth's orbital motion, which can be predicted accurately for millions of years, but which becomes completely unknowable on time scales exceeding τ ∼ 100 Myr [10]. ...
... Note that such routes are subject to modification by new or revised licenses, so this is a snapshot of a fluid picture. Moreover, some routes take multiple paths for some segments, and terminate at several different data centers; where possible we have assumed the shortest path for our estimates.12 Note that this does not necessarily correspond to the order in which the routes have come into operation -indeed, a number are almost certainly still under construction. ...
High frequency trading has led to widespread efforts to reduce information propagation delays between physically distant exchanges. Using relativistically correct millisecond-resolution tick data, we document a 3-millisecond decrease in one-way communication time between the Chicago and New York areas that has occurred from April 27th, 2010 to August 17th, 2012. We attribute the first segment of this decline to the introduction of a latency-optimized fiber optic connection in late 2010. A second phase of latency decrease can be attributed to line-of-sight microwave networks, operating primarily in the 6-11 GHz region of the spectrum, licensed during 2011 and 2012. Using publicly available information, we estimate these networks' latencies and bandwidths. We estimate the total infrastructure and 5-year operations costs associated with these latency improvements to exceed $500 million.
... The celebrated Lorenz system [25] is presented with its essential dynamics as ...
... The maximum Lyapunov dimension obtained in this paper is higher than all other reported values, including two papers based on optimization technique [19,26]. The existing highest Lyapunov dimension is 2.0843, as reported in [19], which is 1.08% higher than the Lyapunov dimension (2.062) of the conventional Lorenz system [25]. However, the maximized Lyapunov dimension in our work is 4.06% higher than the existing highest Lyapunov dimension. ...
The degree of complexity and chaoticness of a chaotic system are, respectively, measured by the Lyapunov dimension and the largest Lyapunov exponent of the system. An increase in these two quantities makes a chaotic system a worthy candidate for different chaos-based applications. This paper provides a generalized methodology to find a set of parameters and initial conditions of any chaotic system that results in the maximum Lyapunov dimension. The proposed approach is validated by maximizing the Lyapunov dimension of the well-known Lorenz system. Competitive swarm optimization is chosen to realize the above objective. The maximum Lyapunov dimension of 2.169 is found for the Lorenz system. Furthermore, in the process of maximization of the Lyapunov dimension, the highest Lyapunov exponent of the Lorenz system is obtained as 7.9138. Thus, the prediction time of the Lorenz system, with the considered parameters and initial conditions, reduces. To the best of our knowledge, our results are the highest among the existing Lyapunov dimension and positive Lyapunov exponent of the Lorenz system. The significance of the increased Lyapunov dimension is demonstrated by using it in an image encryption process. To validate the proposed generalized approach, two more chaotic systems are considered to get higher Lyapunov dimensions. Extensive simulations and analyses are done and presented to substantiate the claims.
... Our approach differs from previous studies since through a formal mathematical argument, we propose a framework in which the reconstructed flow preserves the topology of the underlying dynamics. This is explicitly demonstrated in the classic Lorenz model for finite-amplitude atmospheric convection 33 and by analyzing the time series of one of the variables of the Rössler attractor. 34 ...
... To numerically test our method, we generate a synthetic movie motivated by the classic model developed by Lorenz for an atmospheric convection problem. 33 Considering a layer of fluid of uniform depth H and aspect ratio a, Lorenz proposes a modal decomposition for the stream function (ψ) and the departure of temperature from the non-convective state (θ ). He writes ...
We propose a method based on autoencoders to reconstruct attractors from recorded footage, preserving the topology of the underlying phase space. We provide theoretical support and test the method with (i) footage of the temperature and stream function fields involved in the Lorenz atmospheric convection problem and (ii) a time series obtained by integrating the Rössler equations.
... The classical Lorenz 63 system for ρ = 28 (Lorenz, 1963) is first considered. The relatively simple geometry allows to illustrate some of the phenomena described in Section 2.5. ...
... Appendix A: Lorenz 63 system The Lorenz 63 system is defined by the following equations (Lorenz, 1963) where σ, β and ρ are constant. Usual values are σ = 10, β = 8/3 and ρ = 28. ...
Local dimension computed using Extreme Value Theory (EVT) is usually used as a tool infer dynamical properties of a given state ζ of the chaotic attractor of the system. The dimension computed in this way is also known as the pointwise dimension in dynamical systems literature, and is defined using a limit for infinitely small neighborhood in the phase space around ζ. Since it is numerically impossible to achieve such limit, and because dynamical systems theory predicts that this local dimension is almost constant over the attractor, understanding the properties of this tool for a finite scale R is crucial. We show that the dimension can considerably depend on R, and this view differs from the usual one in geophysics literature, where it is often considered that there is one dimension for a given dynamical state or process. We also systematically assess the reliability of the computed dimension given the number of points to compute it. This interpretation of the R-dependence of the local dimension is illustrated on the Lorenz 63 system for ρ = 28, but also in the intermittent case ρ = 166.5. The latter case shows how the dimension can be used to infer some geometrical properties of the attractor in phase space. The Lorenz 96 system with n = 50 dimensions is also used as a higher dimension example. A dataset of radar images of precipitation (the RADCLIM dataset) is finally considered, with the goal of relating the computed dimension to the (un)stability of a given rain field.
... The goal of this paper is to establish existence and smoothness of the stable foliation for sectional hyperbolic flows. In particular, we treat the case of the classical Lorenz equations [13] ...
... Tucker [19] gave a computer-assisted proof that the classical Lorenz attractor [13] is a robustly transitive invariant set containing an equilibrium. It then follows from [14] that the classical Lorenz attractor is singular hyperbolic. ...
We prove the existence of a contracting invariant topological foliation in a full neighborhood for partially hyperbolic attractors. Under certain bunching conditions it can then be shown that this stable foliation is smooth. Specialising to sectional hyperbolic attractors, we give a verifiable condition for bunching. In particular, we show that the stable foliation for the classical Lorenz equation (and nearby vector fields) is better than which is crucial for recent results on exponential decay of correlations. In fact the foliation is at least .
... Hyperchaos systems have lower stability, making them more resistant to decryption or spectral analysis efforts. Various variants of hyperchaotic systems have been developed to increase the diversity and complexity of key randomization, such as the Lorenz [19], Rössler [20], and Chen et al. [21] systems, which are classic examples of 3D hyperchaotic systems. The 3D Lorenz system is well known for its complex dynamic properties, with some further developments in the Improved Lorenz System [22], which expands the dimension to increase the complexity and security of the encryption. ...
... NPCR calculates the percentage of different pixel values between two encrypted images. Given two encrypted images (C 1 and C 2 ) derived from two slightly different original images, NPCR is defined in Eq. (19). UACI measures the average intensity of differences between C 1 and C 2 . ...
In this paper, we propose a novel secure image communication system that integrates quantum key distribution and hyperchaotic encryption techniques to ensure enhanced security for both key distribution and plaintext encryption. Specifically, we leverage the B92 Quantum Key Distribution (QKD) protocol to secure the distribution of encryption keys, which are further processed through Galois Field (GF(2⁸)) operations for increased security. The encrypted plaintext is secured using a newly developed Hyper 3D Logistic Map (H3LM), a chaotic system that generates complex and unpredictable sequences, thereby ensuring strong confusion and diffusion in the encryption process. This hybrid approach offers a robust defense against quantum and classical cryptographic attacks, combining the advantages of quantum-level key distribution with the unpredictability of hyperchaos-based encryption. The proposed method demonstrates high sensitivity to key changes and resilience to noise, compression, and cropping attacks, ensuring both secure key transmission and robust image encryption.
... The classical Lorenz attractor for the Lorenz equations [24], ...
... A specific example to which our main results apply is the classical Lorenz attractor [24,35] (or more generally, the class of geometric Lorenz attractors [3,16]). More generally still, we consider singular hyperbolic attractors [30]. ...
For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two.
... Information visualization helps analysts detect and examine hidden structure in complex datasets [30]. In particular, few fields have drawn as heavily from visualization as nonlinear dynamics and chaos have for their pivotal discoveries, from Lorenz's first visualization of strange attractors [31], to May's groundbreaking bifurcation diagrams [32], to phase diagrams for discerning higher-dimensional hidden structures in data [33]. Such nonlinear analysis is particularly useful, yet underutilized for exploring time series [34,35]. ...
... Edward Lorenz, the father of chaos theory [38], once described chaos as "when the present determines the future, but the approximate present does not approximately determine the future" [39]. Lorenz first discovered chaos by accident while developing a simple mathematical model of atmospheric convection, using three ordinary differential equations [31]. He found that nearly indistinguishable initial conditions could produce completely divergent outcomes, rendering weather prediction impossible beyond a time horizon of about a fortnight [40]. ...
Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems' behavior.
... We begin our analysis by examining prediction in the classical Lorenz-63 system [20], which exhibits chaotic dynamics. Motivated by the success of the hybrid method in the Lorenz-63 system, we consider a more sophisticated example of predicting the spiking dynamics of a neuron in a network of Hindmarsh-Rose [21] cells. ...
... As a demonstrative example, consider the Lorenz-63 system [20] x = σ(y − x) ...
Scientific analysis often relies on the ability to make accurate predictions of a system's dynamics. Mechanistic models, parameterized by a number of unknown parameters, are often used for this purpose. Accurate estimation of the model state and parameters prior to prediction is necessary, but may be complicated by issues such as noisy data and uncertainty in parameters and initial conditions. At the other end of the spectrum exist nonparametric methods, which rely solely on data to build their predictions. While these nonparametric methods do not require a model of the system, their performance is strongly influenced by the amount and noisiness of the data. In this article, we consider a hybrid approach to modeling and prediction which merges recent advancements in nonparametric analysis with standard parametric methods. The general idea is to replace a subset of a mechanistic model's equations with their corresponding nonparametric representations, resulting in a hybrid modeling and prediction scheme. Overall, we find that this hybrid approach allows for more robust parameter estimation and improved short-term prediction in situations where there is a large uncertainty in model parameters. We demonstrate these advantages in the classical Lorenz-63 chaotic system and in networks of Hindmarsh-Rose neurons before application to experimentally collected structured population data.
... Nonlinear dynamical systems that are irregular, unpredictable and sensitive to initial conditions are categorized as chaotic systems, see for example [21]. In 1963, Lorenz [16] suggested a 3 weather model, which is renowned as the first chaotic attractor. ...
... Differentiating (16) in relation to ε, we get arg( ( )) = 1 ′ 1 − 1 ′ 1 ...
This work focuses on examining the dynamic behavior of a fractional-order hyperchaotic Cai system, a complex mathematical model with potential applications in various fields. The investigation begins by using the Routh-Hurwitz criteria, specifically adapted for fractional-order systems, to evaluate the local asymptotic stability of the equilibrium point. Additionally, the necessary conditions for the occurrence of Hopf bifurcation are derived, highlighting the system's transition from stable equilibrium to oscillatory dynamics. To further illustrate the system's complex behavior, phase portraits and chaotic attractors are generated for various parameter values. These visual representations, supported by detailed numerical simulations, provide a comprehensive depiction of the system's behavior under different conditions. The findings offer valuable insights into the stability and chaotic nature of the fractional-order hyperchaotic Cai system, enriching the understanding of its potential applications in modeling and control of complex systems. Through this analysis, the study contributes to the growing field of fractional-order dynamical systems, emphasizing their ability to describe and analyze phenomena with memory and hereditary properties.
... In this paper, we wish to apply the WL parametrization to a simple dynamical system constructed using as system of interest Lorenz 84 (Lorenz, 1984) and as a forcing system Lorenz 63 (Lorenz, 1963), where the latter influences the evolution of the former thorough a linear coupling and will be appropriately parametrized. This model, already used by (Bódai et al., 2011), will be drastically changed within the paper modifying the value of the time scale separation to switch the roles of slow and fast scale systems between the two models. ...
... Lorenz 63 is probably the most iconic chaotic dynamical system (Saltzman, 1962;Lorenz, 1963;Ott, 1993) and was developed from Navier-Stokes and thermal diffusion equations (see e.g. Hilborn (2000) for a complete, yet simple, derivation of the model) to describe through a simple dynamical system the evolution of three modes corresponding to large scale motions and temperature modulations in the Rayleigh-Bénard convection framework. ...
Constructing accurate, flexible, and efficient parametrizations is one of the great challenges in the numerical modelling of geophysical fluids. We consider here the simple yet paradigmatic case of a Lorenz 84 model forced by a Lorenz 63 model and derive a parametrization using a recently developed statistical mechanical methodology based on the Ruelle response theory. We derive an expression for the deterministic and the stochastic component of the parametrization and we show that the approach allows for dealing seamlessly with the case of the Lorenz 63 being a fast as well as a slow forcing compared to the characteristic time scales of the Lorenz 84 model. We test our results using both standard metrics based on the moments of the variables of interest as well as Wasserstein distance between the projected measure of the original system on the Lorenz 84 model variables and the measure of the parametrized one. By testing our methods on reduced phase spaces obtained by projection, we find support to the idea that comparisons based on the Wasserstein distance might be of relevance in many applications despite the curse of dimensionality.
... To describe how the method can be applied to a flow affected by disturbances, we have chosen the Lorenz system [29], which is one of the best known models in nonlinear dynamics. ...
... As shown by Lorenz [29], a 1D map for the Lorenz system, can be created by taking the consecutive maxima of the variable z. When plotting the pairs (z n , z n+1 ), one gets (approximately) a function f where z n+1 ≈ f (z n ). ...
Transient chaos is a characteristic behavior in nonlinear dynamics where trajectories in a certain region of phase space behave chaotically for a while, before escaping to an external attractor. In some situations the escapes are highly undesirable, so that it would be necessary to avoid such a situation. In this paper we apply a control method known as partial control that allows one to prevent the escapes of the trajectories to the external attractors, keeping the trajectories in the chaotic region forever. To illustrate how the method works, we have chosen the Lorenz system for a choice of parameters where transient chaos appears, as a paradigmatic example in nonlinear dynamics. We analyze three quite different ways to implement the method. First, we apply this method by building a 1D map using the successive maxima of one of the variables. Next, we implement it by building a 2D map through a Poincar\'{e} section. Finally, we built a 3D map, which has the advantage of using a fixed time interval between application of the control, which can be useful for practical applications.
... Chaos theory emerged in the 20th century as researchers began to recognize that deterministic systems could exhibit unpredictable and chaotic behavior. Pioneering work by Lorenz [1] laid the groundwork for this field. In his seminal paper, Lorenz presented a simplified model of atmospheric convection, leading to the famous Lorenz equations. ...
... where σ, r, and b are system parameters [1]. The attractor's distinctive butterfly shape has become a symbol of chaos theory, illustrating the system's sensitive dependence on initial conditions. ...
This study investigates the chaotic dynamics of the Lorentz attractor, a well-known model in chaos theory representing atmospheric convection. The Lorentz system is defined by three coupled nonlinear differential equations that describe the interaction of three variables: x, y, and z. Using the Euler method for numerical integration, we simulate the behavior of the system over a specified time frame, with initial conditions set to x(0) = 1, y(0) = 1, and z(0) = 20.01. The parameters are chosen as σ = 10, b = 8 3 , and r = 28. The results reveal the sensitive dependence on initial conditions characteristic of chaotic systems. The simulations show that as time progresses , x increases steadily, while y and z exhibit a decreasing trend, indicating complex interdependencies among the variables. Notably, the phase space representation highlights the intricate trajectories that reflect the system's chaotic nature. This research contributes to a deeper understanding of chaotic dynamics , illustrating how small variations in initial conditions can lead to vastly different outcomes. The findings have broader implications for fields such as meteorology, engineering, and other areas where chaotic behavior plays a critical role. Through this study, we aim to enhance the comprehension of chaotic systems and their potential applications in real-world scenarios.
... Example 1.5. (Lorenz system) The Lorenz system, introduced in the seminal paper Lorenz (1963), is one of the most famous examples of dynamical systems. Its study has been linked to the development of chaos theory and to the understanding of predictability of atmospheric flows. ...
... A basic definition of predictability will be introduced, based on the concept of Lyapunov exponent. This kind of analysis originated from the seminal work Lorenz (1963) and is based on research work in numerical weather prediction and dynamical system theory. For a more detailed discussion of predictability and chaos theory, see for example, among many others, Boffetta et al (2002), Kalnay (2003), Thompson and Stewart (2002). ...
Lecture notes of a PhD course on numerical methods for Ordinary Differential Equations
... ERA5 is commonly used to drive mesoscale models, providing initial and boundary conditions for higherresolution simulations. Minor differences in initial conditions can lead to drastically different simulation results (Lorenz, 1963), making the accuracy of the input initial conditions crucial. In Section 3, biases in ERA5's temperature and relative humidity under TC conditions are identified, which may result in significant deviations between the simulation results and actual conditions. ...
A detailed analysis of temperature and relative humidity biases in ERA5 reanalysis under tropical cyclone (TC) conditions is conducted using a composite analysis method based on more than 17,000 dropsonde observations. The statistical results indicate that ERA5 underestimates the warm‐core intensity of TCs. It shows a moist bias at the tropopause and a cold‐moist bias in the boundary layer beyond four times the radius of maximum winds (RMW). In contrast, within 4 RMW, a warm‐dry bias in the boundary layer was observed. The temperature bias is more pronounced at the TC center, and relative humidity exhibits a similar pattern, primarily in the mid and upper troposphere. The bias distribution is asymmetric, most prominent in the right front quadrant of the TC's motion direction. Besides, bias correction is performed using an eXtreme Gradient Boosting (XGBoost) model and the biases can be reduced by approximately 60%. The dependence relationships between relative humidity and the influencing factors are investigated through SHapley Additive exPlanations (SHAP) values. The results indicate that low‐level relative humidity increases and instability intensifies in TCs at night. Additionally, the findings suggest that ERA5 may inadequately capture the diurnal variation of relative humidity in TCs. The SHAP analysis also shows that relative humidity tends to be stronger in TC's motion direction.
... Traditional encryption techniques, while effective, often face challenges in balancing security and computational efficiency. To address these challenges, chaos theory shows a new method of proceeding in the current cryptography environment [1,2]. The chaotic method demonstrates an initial stage of random, unpredictable, and erratic conduct, making it ideal for creating complex and secure encryption algorithms. ...
Medical image security is a critical concern in healthcare systems due to the sensitive nature of the data involved. This work presents a scheme that combines cryptographic techniques and other methods to prevent medical images from being compromised. The proposed scheme utilizes the inherent unpredictability of chaotic systems to randomly shuffle image pixels, which significantly improves the diffusion properties of the encryption process. This proposed algorithmic method protects against various types of intruders by saving the given image. Simulation output shows that existing work methods get greater levels of protection, efficiency, and robustness, making them suitable for practical applications in medical data protection. Comprehensive analysis validates the encryption scheme's effectiveness, including key sensitivity, statistical measures, and resistance to common cryptographic attacks, demonstrating its potential as a reliable solution for securing medical images.
... One of the most recognized tenets of chaos theory is the butterfly effect, posited as "the flap of a butterfly's wings in Brazil might set off a tornado in Texas" (Lorenz, 1963). This assertion elucidates an initial event's profound consequences on subsequent outcomes, which analogous to blockchain, says its initial state parameters or consensus are pivotal in determining how it will perform, operate, and ultimately establish its utility. ...
This qualitative grounded theory study investigates the business use cases, existing regulatory policies, and communication governing blockchain in Australia and its practical implementation. The study collected data through anonymous interviews with ten Australian professionals or academics from the blockchain industry, providing in-depth insights and data to develop a grounded theory. Data was collected and codified as part of an ongoing process during the data-gathering phase of the study, enabling the emergence of new theoretical avenues and interesting insights. Opportunities for Australian businesses include enhancing trust, data security, transparency, and the digitalization of value. Moreover, advanced data storage methods, such as off-chain data storage, proof of existence, and zero-knowledge proofs (ZKPs), allow businesses to adopt cutting-edge technological solutions, improving efficiency and security. This study underscores the need for a harmonized regulatory framework, developed through insights from Australian blockchain organizations and the government, to maximize the benefits of this technology and ensure sustainable growth and innovation in Australia’s digital economy. Australia is well-positioned to enhance its global competitiveness in the digital sector by fostering an environment that encourages technological advancements.
... This complexity is often governed by constraints such as geometry, energy flows, material properties, and the interplay between forces. Unlike behavioral complexity, physical complexity tends to exhibit more deterministic behavior, as it is rooted in predictable interactions governed by physics (Carnot, 1824;Timoshenko, 1934;Lorenz, 1963;von Bertalanffy, 1968). ...
Enterprises operate as complex systems embedded in dynamic environments characterized by global interdependencies, technological advancements, and systemic challenges. This paper examines the critical components, challenges, and resilience strategies necessary for modern organizations to navigate complexity and uncertainty. The COVID-19 pandemic and the global semiconductor shortage revealed vulnerabilities in interconnected supply chains, financial markets, and digital ecosystems, underscoring the need for systemic adaptability. Technological advancements, including artificial intelligence, the Internet of Things, and big data, add layers of operational complexity, demanding robust data management, cybersecurity measures, and seamless integration with legacy systems. The paper highlights the growing demand for sustainable and ethical practices, driven by regulatory pressures, consumer expectations, and advocacy group influence. Businesses are compelled to balance short-term efficiency with long-term adaptability to thrive in a volatile environment shaped by rapid technological, market, and societal changes. Case studies of organizations such as Toyota, Amazon, and Unilever illustrate how viewing enterprises as interconnected systems allows them to address root causes of challenges, implement resilience strategies, and leverage adaptability as a competitive advantage. Theoretical frameworks, including systems thinking, complex adaptive systems (CAS), and the Viable System Model, provide tools for understanding enterprise complexity. These frameworks emphasize interdependencies, nonlinearities, feedback loops, and emergent behaviors that define organizational systems. The paper explores the concept of resilience, emphasizing adaptability, recovery, and thriving amidst disruptions as critical elements of long-term sustainability. Challenges such as resistance to change, coordination across subsystems, and the trade-off between efficiency and resilience are analyzed within the context of enterprise architecture, Normal Accident Theory, and the Swiss Cheese Model. The study advocates for adopting resilience engineering and collective mindfulness to anticipate, detect, and manage errors effectively, ensuring organizational stability and growth. By framing enterprises as dynamic, adaptive systems, this paper contributes actionable insights for building resilience, fostering sustainability, and managing complexity in an era of unprecedented disruption.
... This complexity is often governed by constraints such as geometry, energy flows, material properties, and the interplay between forces. Unlike behavioral complexity, physical complexity tends to exhibit more deterministic behavior, as it is rooted in predictable interactions governed by physics (Carnot, 1824;Timoshenko, 1934;Lorenz, 1963;von Bertalanffy, 1968). ...
Enterprises operate as complex systems embedded in dynamic environments characterized by global interdependencies, technol1ogical advancements, and systemic challenges. This paper examines the critical components, challenges, and resilience strategies necessary for modern organizations to navigate complexity and uncertainty. The COVID-19 pandemic and the global semiconductor shortage revealed vulnerabilities in interconnected supply chains, financial markets, and digital ecosystems, underscoring the need for systemic adaptability. Technological advancements, including artificial intelligence, the Internet of Things, and big data, add layers of operational complexity, demanding robust data management, cybersecurity measures, and seamless integration with legacy systems. The paper highlights the growing demand for sustainable and ethical practices, driven by regulatory pressures, consumer expectations, and advocacy group influence. Businesses are compelled to balance short-term efficiency with long-term adaptability to thrive in a volatile environment shaped by rapid technological, market, and societal changes. Case studies of organizations such as Toyota, Amazon, and Unilever illustrate how viewing enterprises as interconnected systems allows them to address root causes of challenges, implement resilience strategies, and leverage adaptability as a competitive advantage. Theoretical frameworks, including systems thinking, complex adaptive systems (CAS), and the Viable System Model, provide tools for understanding enterprise complexity. These frameworks emphasize interdependencies, nonlinearities, feedback loops, and emergent behaviors that define organizational systems. The paper explores the concept of resilience, emphasizing adaptability, recovery, and thriving amidst disruptions as critical elements of long-term sustainability. Challenges such as resistance to change, coordination across subsystems, and the trade-off between efficiency and resilience are analyzed within the context of enterprise architecture, Normal Accident Theory, and the Swiss Cheese Model. The study advocates for adopting resilience engineering and collective mindfulness to anticipate, detect, and manage errors effectively, ensuring organizational stability and growth. By framing enterprises as dynamic, adaptive systems, this paper contributes actionable insights for building resilience, fostering sustainability, and managing complexity in an era of unprecedented disruption.
... Since its modern inception in the pioneering computational work of Charney, Fjörtoft, and Von Neumann (see Charney et al. (1950)), numerical weather prediction (NWP) has proven to present formidable mathematical challenges. In particular, many dynamic models of weather phenomena exhibit multiscale and turbulent features which have been known since the seminal work of Lorenz (1963) to lead to a sensitive dependence on initial conditions. As a consequence, the uncertainties present in a set of initial observations grow exponentially in time under these models, bounding the predictive power of most numerical weather forecasts to medium-range time scales (≤ 14 days). ...
The multiscale and turbulent nature of Earth's atmosphere has historically rendered accurate weather modeling a hard problem. Recently, there has been an explosion of interest surrounding data-driven approaches to weather modeling, which in many cases show improved forecasting accuracy and computational efficiency when compared to traditional methods. However, many of the current data-driven approaches employ highly parameterized neural networks, often resulting in uninterpretable models and limited gains in scientific understanding. In this work, we address the interpretability problem by explicitly discovering partial differential equations governing various weather phenomena, identifying symbolic mathematical models with direct physical interpretations. The purpose of this paper is to demonstrate that, in particular, the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm can learn effective weather models from both simulated and assimilated data. Our approach adapts the standard WSINDy algorithm to work with high-dimensional fluid data of arbitrary spatial dimension. Moreover, we develop an approach for handling terms that are not integrable-by-parts, such as advection operators.
... Although the ensemble mean forecast for lead weeks 4-6 captures the tendency for extended warmer and cooler periods, the forecast overall exhibits large uncertainty and its ensemble mean tends to converge toward climatology. The increase in forecast uncertainty and the ensemble mean's convergence toward climatology is an inherent characteristic of subseasonal weather forecasts, reflecting the intrinsic predictability limit for deterministic weather forecasts of about 2-3 weeks (Domeisen et al., 2018;Lorenz, 1963;Zhang et al., 2019). Predictability on longer timescales is not expected for specific temperature peaks, but subseasonal forecasts can capture tendencies for extended periods of warm or cool temperature anomalies (Vitart et al., 2019;Vitart & Robertson, 2018). ...
Heatwaves pose a range of severe impacts on human health, including an increase in premature mortality. The summers of 2018 and 2022 are two examples with record‐breaking temperatures leading to thousands of heat‐related excess deaths in Europe. Some of the extreme temperatures experienced during these summers were predictable several weeks in advance by subseasonal forecasts. Subseasonal forecasts provide weather predictions from 2 weeks to 2 months ahead, offering advance planning capabilities. Nevertheless, there is only limited assessment of the potential for heat‐health warning systems at a regional level on subseasonal timescales. Here we combine methods of climate epidemiology and subseasonal forecasts to retrospectively predict the 2018 and 2022 heat‐related mortality for the cantons of Zurich and Geneva in Switzerland. The temperature‐mortality association for these cantons is estimated using observed daily temperature and mortality during summers between 1990 and 2017. The temperature‐mortality association is subsequently combined with bias‐corrected subseasonal forecasts at a spatial resolution of 2‐km to predict the daily heat‐related mortality counts of 2018 and 2022. The mortality predictions are compared against the daily heat‐related mortality estimated based on observed temperature during these two summers. Heat‐related mortality peaks occurring for a few days can be accurately predicted up to 2 weeks ahead, while longer periods of heat‐related mortality lasting a few weeks can be anticipated 3 to even 4 weeks ahead. Our findings demonstrate that subseasonal forecasts are a valuable—but yet untapped—tool for potentially issuing warnings for the excess health burden observed during central European summers.
... Great progress has been made in the field of chaos research since Edward Lorenz's pioneering discovery of the first chaotic system in 1963 [16]. In all different fields, such as secure communication, biological systems, image encryption, biomedical engineering, information processing, chemical reactions, economic systems, and so on, the growing importance of chaos systems has been extensively known by researchers [11,12,24]. ...
In this article, the finite time anti-synchronization (FTAS) of master-slave 6D Lorenz systems (MS6DLSS) is discussed. Without using previous study methods, by introducing new study methods, namely by adopting the properties of quadratic inequalities of one variable and utilizing the negative definiteness of the quadratic form of the matrix, two criteria on the FTAS are achieved for the discussed MS6DLSS. Up to now, the existing results on FTAS of chaotic systems have been achieved often by adopting the linear matrix inequality (LMI) method and finite time stability theorems (FTST). Adopting the new study methods studies the FTAS of the MS6DLSS, and the novel results on the FTAS are gotten for the MS6DLSS, which is innovative study work.
... The Lorenz-63 equations have become a standard in chaos theory presentations (Lorenz, 1963) and for data-driven benchmarks (Gauthier et al., 2021;Bollt, 2021;Vlachas et al., 2020). Consideṙ ...
Artificial Neural Networks (ANNs) have proven to be fantastic at a wide range of machine learning tasks, and they have certainly come into their own in all sorts of technologies that are widely consumed today in society as a whole. A basic task of machine learning that neural networks are well suited to is supervised learning, including when learning orbits from time samples of dynamical systems. The usual construct in ANN is to fully train all of the perhaps many millions of parameters that define the network architecture. However, there are certain ANN algorithms that work well with random designs. We have previously presented an explanation as to how the reservoir computing recurrent neural network architecture succeeds despite randomness. Here, we explain how the random feedforward neural networks called the random project networks work. In particular, we present examples for both general function learning and also for learning a flow from samples of orbits of chaotic dynamical systems. There is an interesting geometric explanation of the success, in the case of the ReLu activation function, that relates to the classical mathematical question of how configurations of random lines fall in a plane, or how planes or hyperplanes may fall in higher dimensional spaces. These random configurations lead to a refinement of the domain so that piecewise linear continuous functions result that are dense in continuous functions. This relates neural networks to finite element methods. We highlight the universality of this representation by forecasting the skill of chaotic dynamical systems.
... The theoretical foundation of chaos began with the introduction of the Kolmogorov-Arnold-Moser (KAM) theory of invariant torus in the 1950s-1970s. Subsequently, a series of numerical explorations, such as Lorenz's weather system simulation [3], Hénon and Heiles' study on the existence of the third integral [4], and Sussman and Wisdom's discovery of chaotic motion in Pluto's orbit [5], demonstrated the achievements of chaos dynamics in celestial mechanics. ...
We propose the concept of mutual information for particle pair (MIPP) in curved spacetime, and find that MIPP is a proper chaos indicator. We tested this method in the Kerr spacetime and compared it with the fast Lyapunov indicator. The results show that the MIPP effectively identify orbital states and demonstrates prominent performance in recognizing transitions between orbital states. Our result show that information theory significantly deepen our understanding of dynamics of few-body system.
... As different NN configurations or structures have been constructed for this task, we now evaluate the efficacy of the proposed PINN methodologies utilizing the Lorenz system, which consists of three coupled, nonlinear differential equations and is the common choice when evaluating ODEs [77]. This system is a classic example of chaotic behavior and is described in Eq (17). ...
Physics informed neural networks have been gaining popularity due to their unique ability to incorporate physics laws into data-driven models, ensuring that the predictions are not only consistent with empirical data but also align with domain-specific knowledge in the form of physics equations. The integration of physics principles enables the method to require less data while maintaining the robustness of deep learning in modelling complex dynamical systems. However, current PINN frameworks are not sufficiently mature for real-world ODE systems, especially those with extreme multi-scale behavior such as mosquito population dynamical modelling. In this research, we propose a PINN framework with several improvements for forward and inverse problems for ODE systems with a case study application in modelling the dynamics of mosquito populations. The framework tackles the gradient imbalance and stiff problems posed by mosquito ordinary differential equations. The method offers a simple but effective way to resolve the time causality issue in PINNs by gradually expanding the training time domain until it covers entire domain of interest. As part of a robust evaluation, we conduct experiments using simulated data to evaluate the effectiveness of the approach. Preliminary results indicate that physics-informed machine learning holds significant potential for advancing the study of ecological systems.
... IVT threshold, and the modeling system evaluated. The skill of the prediction is affected by inaccuracies in the initial conditions, approximation in the numerical procedures and physical packages, and the chaotic nature of the atmosphere that limits its predictability (Lorenz 1963). ...
A 200-member ensemble developed at the Center for Western Weather and Water Extremes based on the Weather Research and Forecast atmospheric model tailored for the prediction of atmospheric rivers and associated heavy-to-extreme precipitation events over the Western US (West-WRF) is presented. The ensemble (WW200En) is generated with initial and boundary conditions from the US National Center for Environmental Prediction's Global Ensemble Forecast System (GEFS) and the European Centre for Medium-Range Weather Forecasts' Ensemble Prediction System (EPS), 100 unique combinations of microphysics, planetary boundary layer, and cumulus schemes, as well as perturbations applied to each of the 200 members based on the stochastic kinetic-energy backscatter scheme. Each member is run with 9-km horizontal increments and 60 vertical levels for a 10-month period spanning two winters. The performance of WW200En is compared to GEFS and EPS for probabilistic forecasts of 24-h precipitation, integrated water vapor transport (IVT), and for several thresholds including high percentiles of the observed climatological distribution. The WW200En precipitation forecast skill is better than GEFS at nearly all thresholds and lead times, and comparable or better than the EPS. For larger rainfall thresholds WW200En typically exhibits the best forecast skill. Additionally, WW200En has a better spread-skill relationship than the global systems, and an improved overall reliability and resolution of the probabilistic prediction. The results for IVT are qualitatively similar to those for precipitation forecasts. A sensitivity analysis of the physics parameterizations and the number of ensemble members provides insights into possible future developments of WW200En.
... Σ 1Ẇ1 and Σ 2Ẇ2 are independent white noises multiplied by noise strength matrices. A rich class of turbulent systems belongs to the CGNS family, including the noisy Lorenz 63 system (Lorenz, 1963), the Boussinesq equation, and the rotating shallow water equations, to name a few. Many other systems can systematically be approximated by conditional Gaussian statistical models, which greatly enriches the application of CGNS in fluid dynamics. ...
State estimation in multi-layer turbulent flow fields with only a single layer of partial observation remains a challenging yet practically important task. Applications include inferring the state of the deep ocean by exploiting surface observations. Directly implementing an ensemble Kalman filter based on the full forecast model is usually expensive. One widely used method in practice projects the information of the observed layer to other layers via linear regression. However, when nonlinearity in the highly turbulent flow field becomes dominant, the regression solution will suffer from large uncertainty errors. In this paper, we develop a multi-step nonlinear data assimilation method. A sequence of nonlinear assimilation steps is applied from layer to layer recurrently. Fundamentally different from the traditional linear regression approaches, a conditional Gaussian nonlinear system is adopted as the approximate forecast model to characterize the nonlinear dependence between adjacent layers. The estimated posterior is a Gaussian mixture, which can be highly non-Gaussian. Therefore, the multi-step nonlinear data assimilation method can capture strongly turbulent features, especially intermittency and extreme events, and better quantify the inherent uncertainty. Another notable advantage of the multi-step data assimilation method is that the posterior distribution can be solved using closed-form formulae under the conditional Gaussian framework. Applications to the two-layer quasi-geostrophic system with Lagrangian data assimilation show that the multi-step method outperforms the one-step method with linear stochastic flow models, especially as the tracer number and ensemble size increase.
... Chaos is the term used to describe the nonperiodic behavior of a system. Mathematician and meteorologist Edward Lorenz [1] shaped the understanding of chaotic systems by describing variable evolution through sensitive dependence on initial conditions [2]. In atmospheric science, predictability is defined as the longest time interval to which the accuracy and preciseness of a forecast become no better than the climatological mean [3][4][5]. ...
Tropical cyclone prediction is often described as chaotic and unpredictable on time scales that cross into stochastic regimes. Predictions are bounded by the depth of understanding and the limitations of the physical dynamics that govern them. Slight changes in global atmospheric and oceanic conditions may significantly alter tropical cyclone genesis regions and intensity. The purpose of this paper is to characterize the predictability of seasonal storm characteristics in the North Atlantic basin by utilizing the Largest Lyapunov Exponent and Takens’ Theorem, which is rarely used in weather or climatological analysis. This is conducted for a post-weather satellite era (1960–2022). Based on the accumulated cyclone energy (ACE) time series in the North Atlantic basin, cyclone activity can be described as predictable at certain timescales. Insight and understanding into this coupled non-linear system through an analysis of time delay, embedded dimension, and Lyapunov exponent-reconstructed phase space have provided critical information for the system’s predictability.
... P. L. MCDERMOTT AND C. K. WIKLE Though analog techniques remained known to weather forecasters (e.g., Gringorten, 1955), they lost favor as numerical forecasts became increasingly prevalent with the advent of the digital computer. Yet, Lorenz (1969) considered analogs in evaluating short-term atmospheric predictability, linking the methods to the notions of chaos theory that he discovered (Lorenz, 1963). Although the method was shown to be less useful for short-term forecasting than the numerical weather prediction methods, it was eventually recognized as a viable empirical forecast method for medium to long-range forecasts (e.g., Barnett and Preisendorfer, 1978;van den Dool, 1994) and for climate downscaling (e.g., Zorita et al., 1995;Zorita and Von Storch, 1999). ...
Analog forecasting has been applied in a variety of fields for predicting future states of complex nonlinear systems that require flexible forecasting methods. Past analog methods have almost exclu- sively been used in an empirical framework without the structure of a model-based approach. We propose a Bayesian model framework for analog forecasting, building upon previous analog methods but accounting for parameter uncertainty. Thus, unlike traditional analog forecasting methods, the use of Bayesian modeling allows one to rigorously quantify uncertainty to obtain realistic posterior predictive distributions. The model is applied to the long-lead time forecasting of mid-May averaged soil moisture anomalies in Iowa over a high-resolution grid of spatial locations. Sea Surface Tem- perature (SST) is used to find past time periods with similar trajectories to the current pre-forecast period. The analog model is developed on projection coefficients from a basis expansion of the soil moisture and SST fields. Separate models are constructed for locations falling in each Iowa Crop Reporting District (CRD) and the forecasting ability of the proposed model is compared against a variety of alternative methods and metrics.
... Invariant measures exhibiting Properties (iii) and (iv) are known as physical measures [49]; in such cases the set U is called a basin of µ. Clearly, Properties (i)-(iv) are satisfied if Φ t : X → X is flow on a compact manifold with an ergodic invariant measure supported on the whole of X, but are also satisfied in more general settings, such as certain dissipative flows on noncompact manifolds (e.g., the Lorenz 63 system on X = R 3 [50]). Assuming further that the measures ν S associated with the sampling points y 0 , . . . ...
We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called Vector-valued Spectral Analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data, and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto-Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.
... Singular hyperbolicity is a far-reaching generalization of Smale's notion of Axiom A [51] that allows for the inclusion of equilibria (also known as singular points or steady-states) and incorporates the classical Lorenz attractor [31] as well as the geometric Lorenz attractors of [1,24]. For three-dimensional flows, singular hyperbolic attractors are precisely the ones that are robustly transitive, and they reduce to Axiom A attractors when there are no equilibria [40]. ...
Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the stable foliation for the flow. In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation. These properties include existence of SRB measures, central limit theorems and associated invariance principles, as well as results on mixing and rates of mixing. The properties hold equally for singular hyperbolic flows in higher dimensions provided the center-unstable subspaces are two-dimensional.
... A Lorenz attractor describes patterns of flow (fluid or air) around three "saddle points" in three dimensions (Lorenz, 1963). Individual trajectories circle around one saddle point before flipping over to another, and so on (Fig. 1). ...
Behavioural differences may arise in the absence of genetic or environmental variation. Chaotic dynamics may influence behavioural development, and so this among-individual variation. We discuss methods and experimental designs to test this idea. Ultimately, nonlinear and chaotic behavioural development may explain much of natural variation.
... This feature implies that any small errors in the initial conditions will progressively amplify until the forecast becomes useless, or in other words cannot be distinguished from any random state taken from the climatology of the system. This property was already recognised in the early developments of weather forecasts (Thompson, 1957) and was associated with the nonlinear nature of deterministic dynamical systems by Lorenz (1963). These pioneering works sowed the seeds for the development of predictability theories for the atmosphere and climate, and for important progress in the context of dynamical systems, in particular the development of chaos theory (Eckmann and Ruelle, 1985). ...
The stability properties of intermediate-order climate models are investigated by computing their Lyapunov exponents (LEs). The two models considered are PUMA (Portable University Model of the Atmosphere), a primitive-equation simple general circulation model, and MAOOAM (Modular Arbitrary-Order Ocean-Atmosphere Model), a quasi-geostrophic coupled ocean-atmosphere model on a beta-plane. We wish to investigate the effect of the different levels of filtering on the instabilities and dynamics of the atmospheric flows. Moreover, we assess the impact of the oceanic coupling, the dissipation scheme and the resolution on the spectra of LEs. The PUMA Lyapunov spectrum is computed for two different values of the meridional temperature gradient defining the Newtonian forcing. The increase of the gradient gives rise to a higher baroclinicity and stronger instabilities, corresponding to a larger dimension of the unstable manifold and a larger first LE. The convergence rate of the rate functional for the large deviation law of the finite-time Lyapunov exponents (FTLEs) is fast for all exponents, which can be interpreted as resulting from the absence of a clear-cut atmospheric time-scale separation in such a model. The MAOOAM spectra show that the dominant atmospheric instability is correctly represented even at low resolutions. However, the dynamics of the central manifold, which is mostly associated to the ocean dynamics, is not fully resolved because of its associated long time scales, even at intermediate orders. This paper highlights the need to investigate the natural variability of the atmosphere-ocean coupled dynamics by associating rate of growth and decay of perturbations to the physical modes described using the formalism of the covariant Lyapunov vectors and to consider long integrations in order to disentangle the dynamical processes occurring at all time scales.
... The reasons for choosing these systems are the following: Firstly, all of them are simplified models of well-studied experimental systems. For instance, Lorenz is a simple realization of convective systems (Lorenz, 1963), while the Autocatalator and Rössler have their more complicated analogs in chemical multicomponent reactions (Rössler, 1976;Lynch, 1992). Secondly, chaotic dynamics is extremely nonlinear, highly sensitive, possesses only short-time correlations and is associated with a broad range of frequencies (Guckenheimer and Holmes, 1983;Strogatz, 1994). ...
We have presented a new and alternative algorithm for noise reduction using the methods of discrete wavelet transform and numerical differentiation of the data. In our method the threshold for reducing noise comes out automatically. The algorithm has been applied to three model flow systems - Lorenz, Autocatalator, and Rossler systems - all evolving chaotically. The method is seen to work well for a wide range of noise strengths, even as large as 10% of the signal level. We have also applied the method successfully to noisy time series data obtained from the measurement of pressure fluctuations in a fluidized bed, and also to that obtained by conductivity measurement in a liquid surfactant experiment. In all the illustrations we have been able to observe that there is a clean separation in the frequencies covered by the differentiated signal and white noise.
... Although uniformly hyperbolic attractors are rare, many important properties of hyperbolic systems, including Ruelle's linear response theorem, can also be shown to hold for the far more common non-uniformly hyperbolic or quasi-hyperbolic attractors [17,19,20]. One example of a quasi-hyperbolic attractor is the Lorenz attractor [21]. At the origin of phase space, the Lyapunov covariant vectors for the positive and negative exponent are parallel, so hyperbolicity does not apply. ...
Sensitivity analysis methods are important tools for research and design with simulations. Many important simulations exhibit chaotic dynamics, including scale-resolving turbulent fluid flow simulations. Unfortunately, conventional sensitivity analysis methods are unable to compute useful gradient information for long-time-averaged quantities in chaotic dynamical systems. Sensitivity analysis with least squares shadowing (LSS) can compute useful gradient information for a number of chaotic systems, including simulations of chaotic vortex shedding and homogeneous isotropic turbulence. However, this gradient information comes at a very high computational cost. This paper presents multiple shooting shadowing (MSS), a more computationally efficient shadowing approach than the original LSS approach. Through an analysis of the convergence rate of MSS, it is shown that MSS can have lower memory usage and run time than LSS.
... This approach is hampered by the fact that one can only compute Z up to a finite value of J, limited in size by the available computational power. The Lorenz attractor [12] is defined by ...
We study the asymptotic behavior of the Hopf characteristic function of fractals and chaotic dynamical systems in the limit of large argument. The small argument behavior is determined by the moments, since the characteristic function is defined as their generating function. Less well known is that the large argument behavior is related to the fractal dimension. While this relation has been discussed in the literature, there has been very little in the way of explicit calculation. We attempt to fill this gap, with explicit calculations for the generalized Cantor set and the Lorenz attractor. In the case of the generalized Cantor set, we define a parameter characterizing the asymptotics which we show corresponds exactly to the known fractal dimension. The Hopf characteristic function of the Lorenz attractor is computed numerically, obtaining results which are consistent with Hausdorff or correlation dimension, albeit too crude to distinguish between them.
... As far back as 1952, Turing [49] published pictures of numerical simulations of a nonlinear dynamical model of cell development, exhibiting striking pattern formation. Simulations by Stein & Ulam [44,45] and Lorenz [24] gave persuasive pictorial evidence of complicated structure in attractors, but attracted little attention when they were published. Hamming's review [15] of [45] was unenthusiastic: ...
Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that no directed feedback loops are negative. Several new results for cooperative systems are obtained in the process.
... These ODEs are obtained by starting with the familiar Lorenz system [15]:ẋ = −σ(x − y),ẏ = −xz + ρx − y,ż = xy − βz, letting σ → ∞, and rescaling the variables and the remaining parameters. Lorenz fixed the value of a at 0.36, and varied the time step τ . ...
In a one-parameter study of a noninvertible family of maps of the plane arising in the context of a numerical integration scheme, Lorenz studied a sequence of transitions from an attracting fixed point to "computational chaos." As part of the transition sequence, he proposed the following as a possible scenario for the breakup of an invariant circle: the invariant circle develops regions of increasingly sharper curvature until at a critical parameter value it develops cusps; beyond this parameter value, the invariant circle fails to persist, and the system exhibits chaotic behavior on an invariant set with loops [Lorenz, 1989]. We investigate this problem in more detail and show that the invariant circle is actually destroyed in a global bifurcation before it has a chance to develop cusps. Instead, the global unstable manifolds of saddle-type periodic points are the objects which develop cusps and subsequently "loops" or "antennae." The one-parameter study is better understood when embedded in the full two-parameter space and viewed in the context of the two-parameter Arnold horn structure. Certain elements of the interplay of noninvertibility with this structure, the associated invariant circles, periodic points and global bifurcations are examined.
... We remark that applying Theorem 1.2 to concrete systems, one can get some necessary conditions on elementary integrability of the systems. For example, the Lorenz system [16] (1.2)ẋ = s(y − x),ẏ = rx − y − xz,ż = −bz + xy, with s, r, b parameters, has the six irreducible invariant algebraic surfaces (Darboux polynomials in terms of the terminology from the Darboux theory of integrability): ...
Prelle and Singer showed in 1983 that if a system of ordinary differential equations defined on a differential field K has a first integral in an elementrary field extension L of K, then it must have a first integral consisting of algebraic elements over K via their constant powers and logarithms. Based on this result they further proved that an elementary integrable planar polynomial differential system has an integrating factor which is a fractional power of a rational function. Here we extend their results and prove that any n dimensional elementary integrable polynomial vector field has functionally independent first integrals being composed of algebraic elements over K. Furthermore, using the Galois theory we prove that the vector field has a rational Jacobian multiplier.
... The third example tested is the Lorenz attractor ODE system [24] that is commonly used as a benchmark ODE for similar works on the analysis of dynamical models due to its characteristic chaotic behavior that occurs with certain combinations of model parameter values [25]. The Lorenz system is a foundational dynamical model in the areas of chaos theory and weather modeling such as atmospheric convection [26]. ...
This work focused on demonstrating the use of dynamic time warping (DTW) as a metric for the elementary effects computation in Morris-based global sensitivity analysis (GSA) of model parameters in multivariate dynamical systems. One of the challenges of GSA on multivariate time-dependent dynamics is the modeling of parameter perturbation effects propagated to all model outputs while capturing time-dependent patterns. The study establishes and demonstrates the use of DTW as a metric of elementary effects across the time domain and the multivariate output domain, which are all aggregated together via the DTW cost function into a single metric value. Unlike the commonly studied coefficient-based functional approximation and covariance decomposition methods, this new DTW-based Morris GSA algorithm implements curve alignment via dynamic programing for cost computation in every parameter perturbation trajectory, which captures the essence of “elementary effect” in the original Morris formulation. This new algorithm eliminates approximations and assumptions about the model outputs while achieving the objective of capturing perturbations across time and the array of model outputs. The technique was demonstrated using an ordinary differential equation (ODE) system of mixed-order adsorption kinetics, Monod-type microbial kinetics, and the Lorenz attractor for chaotic solutions. DTW as a Morris-based GSA metric enables the modeling of parameter sensitivity effects on the entire array of model output variables evolving in the time domain, resulting in parameter rankings attributed to the entire model dynamics.
В статье рассмотрен принцип наименьшего действия электротехнической системы в режимах детерминированного хаоса. Представлена модель Э. Лоренца. Разработана модель трехфазного источника электротехнической системы, в котором возникает детерминированный хаос. Составлено выражение определения действия для хаотических колебаний. Разработана физическая модель, демонстрирующая хаотические колебания в электротехнической системе. Представлена функция численного метода Рунге- Кутте 4-го порядка. Построены диаграммы хаотических колебаний и фазовые портреты. Приведена диаграмма действия S от времени для системы, которая испытывает хаотические колебания. Выдвинута гипотеза о возможности определения перехода электротехнической системы в хаотический режим с помощью величин амплитуды, фазы и формы действия S. Сделано предположение о том, что с помощью параметров амплитуды фазы и формы действия S появляется возможность определения перехода электротехнической системы в хаотический режим.
The article discusses the principle of the least action of an electrical system in modes of deterministic chaos. The model of E. Lorenz is presented. A model of a three-phase source of an electrical system in which deterministic chaos arises has been developed. An expression for the definition of action for chaotic oscillations has been compiled. A physical model has been developed that demonstrates chaotic oscillations in an electrical system. The function of the 4th order numerical Runge-Kutte method is presented. Diagrams of chaotic oscillations and phase portraits were constructed. A diagram of the action S versus time is given for a system that experiences chaotic oscillations. A hypothesis has been put forward about the possibility of determining the transition of an electrical system to a chaotic mode using the values of the amplitude, phase and form of action S. It is assumed that using the parameters of the amplitude of the phase and form of action S it becomes possible to determine the transition of an electrical system to a chaotic mode.
Лаконичные ориентирующие материалы студентам и аспирантам технического университета к темам системного мышления, развитию творческих способностей. Компактное учебное пособие содержит тексты блиц-докладов автора и ссылки на их видеозаписи, образуя удобный для восприятия учебный материал. Рекомендуется в качестве дополнения к университетским лекциям и занятиям или как самостоятельное краткое пособие, конспект тематических лекций. Также рекомендуется всем, кто интересуется формированием навыков научной работы.
The low-frequency variability of the mid-latitude atmosphere involves complex nonlinear and chaotic dynamical processes posing predictability challenges. It is characterized by sporadically recurring, often long-lived patterns of atmospheric circulation of hemispheric scale known as weather regimes. The evolution of these circulation regimes in addition to their link to large-scale teleconnections can help extend the limits of atmospheric predictability. They also play a key role in sub- and inter-seasonal weather forecasting. Their identification and modeling remains an issue, however, due to their intricacy, including a clear conceptual picture. In recent years, the concept of metastability has been developed to explain regimes formation. This suggests an interpretation of circulation regimes as communities of states in which the atmospheric system remains in their neighborhood for abnormally longer than typical baroclinic timescales. Here we develop a new and effective method to identify such communities by constructing and analyzing an operator of the system's evolution via hidden Markov model (HMM). The method makes use of graph theory and is based on probabilistic approach to partition the HMM transition matrix into weakly interacting blocks -- communities of hidden states -- associated with regimes. The approach involves nonlinear kernel principal component mapping to consistently embed the system state space for HMM building. Application to northern winter hemisphere using geopotential heights from reanalysis yields four persistent and recurrent circulation regimes. Statistical and dynamical characteristics of these circulation regimes and surface impacts are discussed. In particular, unexpected high correlations are obtained with EL-Nino Southern Oscillation and Pacific decadal oscillation with lead times of up to one year.
Ensemble forecasts are frequently utilized to assess the uncertainties of prediction systems. There is a consensus that generating initial perturbations with specific structures is more conducive to characterizing the growth dynamics of analysis errors, demonstrating higher forecast skills. However, the widely used methods, such as linear singular vectors (SVs) and orthogonal conditional nonlinear optimal perturbations (O‐CNOPs) exhibit strong linear‐assumption dependence and the overestimation of growing properties for analysis errors, respectively, severely limiting the ability to capture analysis errors and forecast skills. To tackle these challenges, a theoretical framework is established to solve the optimal distribution of the nonlinear growing‐type initial perturbations by variational inference (VI) incorporated with the concept of CNOPs, marked as VI‐CNOPs. As the distribution is obtained, diverse initial perturbations for ensemble forecasts can be easily sampled in it. To evaluate the reliability of VI‐CNOPs, a series of ensemble forecast experiments are then conducted using the Lorenz‐96 model. We compare the deterministic and probabilistic forecast skills of VI‐CNOPs, O‐CNOPs, and SVs under various optimization durations. The results reveal that, as the optimization durations extend, the forecast skills of VI‐CNOPs progressively improve, consistently outperforming O‐CNOPs and SVs. This trend remains consistent across various forecast lead times. Further analysis reveals that VI‐CNOPs more effectively capture the covariance matrix of analysis errors, aligning with the fundamental concept of perturbation generation methods for ensemble forecasts. Moreover, unlike O‐CNOPs and SVs, VI‐CNOPs do not require the utilization of adjoint and tangent linear models, largely expanding its application. These results indicate the novelty and efficacy of VI‐CNOPs for ensemble forecasts.
Standard methods of data assimilation assume prior knowledge of a model that describes the system dynamics and an observation function that maps the model state to a predicted output. An accurate mapping from model state to observation space is crucial in filtering schemes when adjusting the estimate of the system state during the filter's analysis step. However, in many applications the true observation function may be unknown and the available observation model may have significant errors, resulting in a suboptimal state estimate. We propose a method for observation model error correction within the filtering framework. The procedure involves an alternating minimization algorithm used to iteratively update a given observation function to increase consistency with the model and prior observations, using ideas from attractor reconstruction. The method is demonstrated on the Lorenz 1963 and Lorenz 1996 models, and on a single-column radiative transfer model with multicloud parameterization.
Synchronization is the process of achieving identical dynamics among coupled identical units. If the units are different from each other, their dynamics cannot become identical; yet, after transients, there may emerge a functional relationship between them -- a phenomenon termed "generalized synchronization." Here, we show that the concept of transient uncoupling, recently introduced for synchronizing identical units, also supports generalized synchronization among nonidentical chaotic units. Generalized synchronization can be achieved by transient uncoupling even when it is impossible by regular coupling. We furthermore demonstrate that transient uncoupling stabilizes synchronization in the presence of common noise. Transient uncoupling works best if the units stay uncoupled whenever the driven orbit visits regions that are locally diverging in its phase space. Thus, to select a favorable uncoupling region, we propose an intuitive method that measures the local divergence at the phase points of the driven unit's trajectory by linearizing the flow and subsequently suppresses the divergence by uncoupling.
This study examines the dynamical properties of the Ikeda map, with a focus on bifurcations and chaotic behavior. We investigate how variations in dissipation parameters influence the system, uncovering shrimp-shaped structures that represent intricate transitions between regular and chaotic dynamics. Key findings include the analysis of period-doubling bifurcations and the onset of chaos. We utilize Lyapunov exponents to distinguish between stable and chaotic regions. These insights contribute to a deeper understanding of nonlinear and chaotic dynamics in optical systems.
The main goal of the paper is to present recent results obtained in the mathematical theory of dynamical chaos and related to the discovery of its new, third, form, the so-called mixed dynamics. This type of chaos is very different from its two classical forms, conservative and dissipative chaos, and its main difference is that attractors and repellers can intersect without coinciding. The main results of the paper are related to construction of theoretical schemes aimed to mathematical justification of this phenomenon using the most general methods of topological dynamics. The paper also provides a number of examples of systems from applications in which mixed dynamics is observed. It is shown that such dynamics can be of different types, from close to conservative to strongly dissipative, and also that it can arise as a result of various bifurcation mechanisms.
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