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Numerical methods can suppress chaos
Abstract
Numerical methods for the solution of ordinary differential equations are one of the main tools used in the theoretical investigation of nonlinear continuous dynamical systems. These replace the continuous dynamical system under study by a discrete dynamical system that is then usually simulated on a digital computer. It is well known that such discrete dynamical systems may be chaotic even when the underlying continuous dynamical system is not chaotic. We here show that some numerical methods may produce discrete dynamical systems that are not chaotic, even when the underlying continuous dynamical system is thought to be chaotic. We find in this case that the transition to chaos from false stability mimics the transition to chaos that has been previously observed as parameters were changed in the Rössler system.
... These conditions can be seen as errors added to a system when simulated in the computational domain. Some works have shown how to control these errors [4] [5]. ...
... Figure 2 is the comparison between Case 1 and Case 2, where we altered the Runge-Kutta notation. Therefore, it shows the voltage across capacitor C1, using the formulation expressed in Equation (1) and considering interval extensions shown in Equation (5) and Equation (6). Figure 3 depicts the comparison between Case 1 and Case 3, where we altered the differential equation for the voltage in capacitor C1. Therefore, it shows the voltage across capacitor C1, using the formulation expressed in Equation (1) and Equation (7) and considering Runge-Kutta notation shown in Equation (5). ...
... Therefore, it shows the voltage across capacitor C1, using the formulation expressed in Equation (1) and considering interval extensions shown in Equation (5) and Equation (6). Figure 3 depicts the comparison between Case 1 and Case 3, where we altered the differential equation for the voltage in capacitor C1. Therefore, it shows the voltage across capacitor C1, using the formulation expressed in Equation (1) and Equation (7) and considering Runge-Kutta notation shown in Equation (5). Despite the differences, the results highlight that a small change in the model led to a change in the results without significantly alter the accuracy. ...
... Instead, as explained for instance in [6], small backward error can be perfectly satisfactory as an explanation of the success of a numerical method on a chaotic problem. Interestingly, numerical methods can suppress true chaos, but only if the backward error is large [7]. ...
... This means that even though the solution will decay, it must be the solution of a problem that is more than 100% different to the original problem. Likewise, a substantial portion of the right-half plane will have |δ| > 1 for implicit Euler; this is why such numerical methods can suppress (actual) chaos [7]. In contrast, the implicit midpoint rule (with θ = 1/2) does not have such large 100% error zones (although it does, near the portions of the real axis where |μ| > 2). ...
... Going from(7) to(8) is correct over the positive reals but not quite right over C: ln z means the principal branch of the logarithm, with argument in (−π, π]. We will see how to choose the correct complex branch in(10), in a later section. ...
We show how to compute the optimal relative backward error for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of a general approach that uses results from optimal control theory to compute optimal residuals, but elementary methods can also be used here because the problem is so simple. This analysis produces some new insights into the numerical solution of stiff problems.
The optimal backward error for the Dahlquist test problem when the midpoint method is used. Computed via a Padé approximant.
... Thus, it is necessary and essential to construct numerical schemes to guarantee the positivity and stability of the solution. However, numerical schemes such as Euler and Runge-Kutta methods can generate oscillations, bifurcations, chaos, and false steady states [15,16]. Enatsu et al. [17] pointed out that how to choose the discrete schemes that preserve the global asymptotic stability for equilibria of the corresponding continuous-time epidemic models was still an open problem. ...
... N B is the Laplacian form of the matrix . Ň ij / [15]. The irreducibility of B implies that matrices . ...
... Let C kj be the cofactor of the .k, j/ entry of N B. By Theorem A in [15], system N B D 0 has a positive solution D . 1 , 2 , , m / where k D C kk > 0 for k D 1, 2, , m. Notice that the Volterra functionˆ. ...
In this paper, by applying nonstandard finite difference scheme, we propose a discrete multigroup Susceptible-Infective-Removed (SIR) model with nonlinear incidence rate. Using Lyapunov functions, it is shown that the global dynamics of this model are completely determined by the basic reproduction number . If , then the disease-free equilibrium is globally asymptotically stable; if , then there exists a unique endemic equilibrium and it is globally asymptotically stable. Example and numerical simulations are presented to illustrate the results. Copyright
... The initial conditions for system (2) are ...
... The obtained discrete model should preserve the dynamical properties of the corresponding continuous model as much as possible. However, traditional numerical schemes such as Euler and Runge-Kutta sometimes fail generating oscillations, bifurcations, chaos and false steady states [2,6]. In order to prevent such numerical instability, the non-standard finite difference (NSFD) scheme proposed by Mickens [15][16][17] is well known and applied to various continuous epidemic models with or without time delay [5,8,10,23,24,31] and references therein. ...
... The main aim of this paper is to show that the discrete system (6) which derived by using Mickens' scheme can efficiently preserves the global asymptotic stability of the equilibria for original continuous system (2). The rest of this paper is organized as follows. ...
In this paper, we designed and analysed a discrete model to solve a delayed within-host viral infection model by using non-standard finite difference scheme. The original model that we considered was a delayed viral infection model with cell-to-cell transmission, cell-mediated immune response and general nonlinear incidence. We show that the discrete model has equilibria which are exactly the same as those of the original continuous model and the conditions for those equilibria to be globally asymptotically stable are consistent with the original continuous model with no restriction on the time step size. The results imply that the discretization scheme can efficiently preserves the qualitative properties of solutions for corresponding continuous model.
... Conversely, Lorenz (1989) gives examples of chaotic behavior which occurs when difference equations, used as approximations to ODEs, are solved numerically with an 'excessively' large time step. Corless et al. (1991) also show that floating-point simulation of a discrete map with a globally attracting fixed point at the origin can appear chaotic, for extremely long time, purely due to rounding error effects. An interesting review on numerical problems for chaotic dynamical systems is Corless (1994). ...
... Moreover, it has been shown that numerical experiments can sometimes yield dynamics which are qualitatively completely different from the true ones. For instance, Corless et al. (1991) show that the implicit Euler method can, for large enough step size, artificially stabilize truly unstable fixed points and completely destroy any possible chaotic attractors. Conversely, Lorenz (1989) gives examples of chaotic behavior which occurs when difference equations, used as approximations to ODEs, are solved numerically with an 'excessively' large time step. ...
... it Euler method can, for large enough step size, artificially stabilize truly unstable fixed points and completely destroy any possible chaotic attractors. Conversely, Lorenz (1989) gives examples of chaotic behavior which occurs when difference equations, used as approximations to ODEs, are solved numerically with an 'excessively' large time step. Corless et al. (1991) also show that floating-point simulation of a discrete map with a globally attracting fixed point at the origin can appear chaotic, for extremely long time, purely due to rounding error effects. An interesting review on numerical problems for chaotic dynamical systems is Corless (1994). Therefore, a fundamental question arises: under wh ...
In the field of chaotic time series analysis, there is a lack of a distributional theory for the main quantities used to characterize the underlying data generating process (DGP). In this paper a method for resampling time series generated by a chaotic dynamical system is proposed. The basic idea is to develop an algorithm for building trajectories which lie on the same attractor of the true DGP, that is with the same dynamical and geometrical properties of the original data. We performed some numerical experiments on some short noise-free and high-noise series confirming that we are able to correctly reproduce the distribution of the largest finite-time Lyapunov exponent and of the correlation dimension.
... 2.4). And since the set of points encoded by a computer is necessarily finite, by solving these systems through numerical simulations the possibility of chaos is eliminated [9], and all computed orbits are periodic [8]. The study of finite precision effects in an algorithm is, therefore, vital [17]. ...
... As explained in [2, §3], if each real value cannot be given a representation in a code and only a finite set of them is encoded, then, with few exceptions, an orbit generated by a recursive system cannot be calculated in that code and its period cannot be calculated either. In that case, only pseudo-orbits [16], [25], [26] and their pseudo-periods [9], [8] can be obtained. Each new calculated point is simply an approximate point. ...
... where aES denotes the differential gain of ES, ξ represents the gain limiting factor, VB is the total volume of QDs, and VS denotes the intra-cavity laser field volume. Numerical methods for the solution of ordinary differential equations are the main tools to investigate the nonlinear dynamical systems [40,41]. In this work, a desktop PC with a six-core processor (AMD Ryzen 5 1600X, Advanced Micro Devices Inc., Santa Clara, CA, USA) and 16GB installed memory is used to perform the simulation, and we adopt the ode45 solver (Fourth-Fifth order Runge-Kutta algorithm, where the fourth-order provides the candidate solutions and the fifth-order controls the errors) in MATLAB software to solve the above differential equations, after taking into account the accuracy and speed of the calculations. ...
... In this work, a desktop PC with a six-core processor (AMD Ryzen 5 1600X, Advanced Micro Devices Inc., Santa Clara, CA, USA) and 16GB installed memory is used to perform the simulation, and we adopt the ode45 solver (Fourth-Fifth order Runge-Kutta algorithm, where the fourth-order provides the candidate solutions and the fifth-order controls the errors) in MATLAB software to solve the above differential equations, after taking into account the accuracy and speed of the calculations. Since the step size will affect the simulation results [40], we use the adaptive step size in numerical simulations. The used parameters for the ES-QD laser during the simulations are given in Table 1 [33,38]. ...
We numerically investigate the nonlinear dynamic properties of an exclusive excited-state (ES) emission quantum dot (QD) laser under optical injection. The results show that, under suitable injection parameters, the ES-QD laser can exhibit rich nonlinear dynamical behaviors, such as injection locking (IL), period one (P1), period two (P2), multi-period (MP), and chaotic pulsation (CP). Through mapping these dynamic states in the parameter space of the frequency detuning and the injection coefficient, it can be found that the IL occupies a wide region and the dynamic evolution routes appear in multiple forms. Via permutation entropy (PE) calculation to quantify the complexity of the CP state, the parameter range for acquiring the chaos with high complexity can be determined. Moreover, the influence of the linewidth enhancement factor (LEF) on the dynamical state of the ES-QD laser is analyzed. With the increase of the LEF value, the chaotic area shrinks (expands) in the negative (positive) frequency detuning region, and the IL region gradually shifts towards the negative frequency detuning.
... It is known [66] that discrete maps of continuous chaotic systems may exhibit characteristic phase volume deformations due to discretization effects during long-term computer simulations. Recently we have shown that by adjusting the discrete integration operator properties various aspects of the system dynamics can be effectively controlled, including the phase volume and shape governing also the inherent symmetry laws [67]. ...
... (b) The x-y projection of the attractor, where ''Regime A'' trajectory corresponds to the first steps of the explicit midpoint algorithm , while ''Regime B'' trajectory is obtained after considerable simulation time. (c) The ''Regime C'' trajectory was obtained by the implicit midpoint algorithm and demonstrates the chaos suppression effect[66]. ...
... Traditional numerical schemes such as Euler's method and Runge-Kutta method, however, may induce numerical instability, as the discretization would change properties of solutions, such as stability and positivity, of the original model, thus numerical scheme should be carefully chosen to preserve nature of the original system, see e.g. [4,8]. ...
... In the ultra discrete model (9) time delay does not change qualitative dynamics, but changes the solution behavior. 4. Monotone convergence in a two-dimensional lattice. ...
We propose an ultra-discretization for an SIR epidemic model with time delay. It is proven that the ultra-discrete model has a threshold property concerning global attractivity of equilibria as shown in differential and difference equation models. We also study an interesting convergence pattern of the solution, which is illustrated in a two-dimensional lattice.
... Integration of the nonlinear partial differential equation is performed over long time intervals (up to several thousand cycles) so that any existing chaotic behavior is revealed, therefore it is of great advantage to use the largest time step possible without introducing spurious solutions. It has been established that numerical schemes can both suppress existing chaos (Corless (1991)) and introduce non-existing chaos (Lorenz (1989); Reinhall et al. (1989)). Examples are Hirai and Adachi (1994) who show chaotic behavior in the Runge-Kutta simulation of a nonlinear differential equation although no chaos exists in the original differential equation. ...
... Reinhall et al. (1989) show that in the study of a Hamiltonian Duffing oscillator chaotic solutions are caused by using Euler's method of numerical integration. Tongue (1987), Lorenz (1989), Yee et al. (1991), Yee and Sweby (1992); Corless et al. (1991), Corless (1992) and Humphries (1993) provide a deeper insight into numerical chaos. ...
The time spectral element (TSE) method is a high order accurate method for solving nonlinear and chaotic temporal dynamical systems. It's performance on the Hamiltonian Duffing equation and a bilinear oscillator is examined and compared with standard numerical schemes. These systems were chosen for analysis due to the availability of accurate solutions independent of spurious discretization effects. The implicit form of the TSE, is unconditionally stable in the linear case. For both systems p-convergence is exponential and the h-convergence rate of the end points is of order 2p≦α≦2p+1, (where α is the convergence rate and p the polynomial order). The explicit form of the TSE is conditionally stable for the linear system. The TSE method competes successfully with the other numerical schemes examined where high accuracy is desired and therefore proves an attractive numerical method for simulation of chaotic dynamical systems.
... However, digital implementations of chaotic systems would become periodic eventually due to finite precisions. Furthermore, temporal discretization would result with the harmful effect in the higher-dimensional chaotic systems [27]. These could introduce security risks in chaos-based cryptosystems. ...
... In contrast to the encryption techniques [8][9][10][11][12]26,28,29], the NPCR, UACI values in this paper are more close to the ideal values. And the algorithms [11,28] have been deciphered by the chosen or the known-plaintext attacks [27,31]. It can be seen from Tables 8, 9, 10, 11 that the image encryption algorithms [26] and [28] are more robust against cropping, histogram equalization and contrast adjustment attacks than the proposed image encryption scheme. ...
An image encryption scheme is proposed using high-dimensional chaotic systems and cycle operation for DNA sequences. In the scheme, the pixels of the original image are encoded randomly with the DNA coding rule controlled by a key stream produced from Chen’s hyper-chaos. In addition to confusion on the DNA sequence matrix with Lorenz system, a cycle operation for DNA sequences is projected to diffuse the pixel values of the image. In order to enhance the diffusion effect, a bitwise exclusive-OR operation is carried out for the decoded matrices with a binary key stream, and then the cipher-image is obtained. Simulation results demonstrate that the proposed image encryption scheme with acceptable robustness is secure against exhaustive attack, statistical attack and differential attack.
... Let us consider the case of systems sensitive to initial conditions within continuous mechanics. It is known that the discretization of chaotic differential equations can suppress the chaotic behaviour of the system (Corless at al. 1991). 20 In other words, the system described with differential equations has not the same general behaviour than the one described with difference equations. ...
... 21 This property is not so surprising. Corless et al. (1991) emphasize that the suppression of chaotic behaviour due to discretization occurs if time steps are taken too large. However, what was not expected is that the Poincaré sections are actually more reliable within DM rather within continuous mechanics in using traditional numerical integrators (Rowley & Marsden 2002). ...
In this paper, I present a discrete solution for the paradox of Achilles and the tortoise. I argue that Achilles overtakes the tortoise after a finite number of steps of Zeno’s argument if time is represented as discrete. I then answer two objections that could be made against this solution. First, I argue that the discrete solution is not an ad hoc solution. It is embedded in a discrete formulation of classical mechanics. Second, I show that the discrete solution cannot be falsified experimentally.
... However, in addition to spatial discretization, they require an integration method for digital realizations. Although the shadowing lemma is used to justify the numerical simulations of the differential equation-based chaotic systems, spatial discretization tends to stabilize the orbits, and poor integration methods may introduce unexpected behaviors [9,11]. Again, applying periodic perturbations, in other terms, injecting a small noise to the system improves the results and prevents the stabilization effect by destroying short periodic orbits [6,11]. ...
... As a result, the system response may quickly become unrelated to the original system response [8]. For example, the implicit Euler integration method stabilizes truly unstable fixed points and destroys any possible chaotic attractors in [9]. Therefore, numerical integration results should be kept as near as possible to the original system behavior [8]. ...
In this paper, a novel method is proposed for producing pseudo random numbers from differential equation-based chaotic systems. The proposed method utilizes a Lü-like chaotic system capable of exhibiting both Lorenz-like and Chen-like chaotic system behaviors for different parameter values. A parameter switching process controlled by a linear feedback shift register is used to obtain the transition between Lorenz-like and Chen-like behaviors. The system has a new initial condition for each transition. Thus, both the stabilization effect and the propagation of the integration errors that occur in the discretization of the differential equation-based chaotic systems are prevented. Additionally, the more complex behavior of the pseudo orbits in the phase plane makes future predictions difficult. The pseudo random numbers produced by the proposed method are tested with NIST SP 800-22 and TestU01 randomness test suites, and all tests are passed. The field programmable gate array implementation of the pseudo random number generator based on the proposed method is also presented, and the implementation results are compared with the implementation of a chaotic map-based pseudo random number generator.
... However, the truth is that rounding errors affect numerical simulations of dynamical systems in very complex ways [7]. Well-conditioned dynamical systems may display chaotic numerical behavior [8] [9]. Conversely, numerical methods can suppress chaos in some chaotic dynamical systems [9]. ...
... Well-conditioned dynamical systems may display chaotic numerical behavior [8] [9]. Conversely, numerical methods can suppress chaos in some chaotic dynamical systems [9]. ...
We propose the use of affine arithmetic in cell-mapping methods for the robust visualization of strange attractors and show that the resulting cellular approximations converge faster than those produced by cell-mapping methods based on classical interval arithmetic.
... Integration of the nonlinear partial differential equation is performed over long time intervals (up to several thousand cycles) so that any existing chaotic behavior is revealed, therefore it is of great advantage to use the largest time step possible without introducing spurious solutions. It has been established that numerical schemes can both suppress existing chaos (Corless (1991)) and introduce non-existing chaos (Lorenz (1989); Reinhall et al. (1989)). Examples are Hirai and Adachi (1994) who show chaotic behavior in the Runge-Kutta simulation of a nonlinear differential equation although no chaos exists in the original differential equation. ...
... Reinhall et al. (1989) show that in the study of a Hamiltonian Duffing oscillator chaotic solutions are caused by using Euler's method of numerical integration. Tongue (1987), Lorenz (1989), Yee et al. (1991), Yee and Sweby (1992); Corless et al. (1991), Corless (1992) and Humphries (1993) provide a deeper insight into numerical chaos. ...
The time spectral element (TSE) method is a high order accurate method for solving nonlinear and chaotic temporal dynamical systems. It's performance on the Hamiltonian Duffing equation and a bilinear oscillator is examined and compared with standard numerical schemes. These systems were chosen for analysis due to the availability of accurate solutions independent of spurious discretization effects. The implicit form of the TSE, is unconditionally stable in the linear case. For both systems p-convergence is exponential and the h-convergence rate of the end points is of order 2p≦α≦2p+1, (where α is the convergence rate and p the polynomial order).
The explicit form of the TSE is conditionally stable for the linear system. The TSE method competes successfully with the other numerical schemes examined where high accuracy is desired and therefore proves an attractive numerical method for simulation of chaotic dynamical systems.
... Often, chaotic systems are simulated using traditional fixed-step approaches such as explicit Euler, implicit Euler or Runge-Kutta (R-K) often of order 4. However, there are many examples in literature evidencing instabilities such as chaos suppression or induction when using fixed-step methods on chaotic systems [2], [3,4,5]. In general, the misbehavior is associated to the choice of the integration step size [5]. ...
... Among those methods, spatial encryption methods based on the chaos system have attracted extensive concerns for the close connection between chaos and encryption. However, discrete chaotic systems will become periodic eventually and can cause a harmful effect in higher dimensional chaotic systems (Wheeler et al., 1989;Corless et al., 1991). To overcome the short period issue existing in discrete chaotic system, spatiotemporal chaotic system with longer period such as Logistic-Sine system (LSS) in the coupled map lattice (CML) (Wang et al., 2015), Mixed Linear Nonlinear Coupled Map Lattices (MLNCML) system (Tao et al., 2007). ...
In this paper, an angular graph Fourier transform (AGFT) is introduced to encrypt color images with their intrinsic structures. The graph Fourier transform (GFT) is extended to the AGFT and proven to have the desired properties of angular transform and graph transform. In the proposed encryption method, color images are encoded by DNA sequences and confused under the control of chaotic key streams firstly. Secondly, sparse decomposition based on the random walk is applied to scramble pixels spatially, and a series of sub-images are obtained. This step increases encryption efficiency. Finally, the intrinsic sub-image structure is reflected by graphs, and the signals on different subgraphs are transformed into different AGFT domains with particular angular parameters, which makes the proposed method relevant to the original image structure and enhances security. The experimental results demonstrate that the proposed algorithm can resist various potential attacks and achieve better performance than the state-of-the-art algorithms.
... Such systems are not always easy to solve analytically and are often studied through robust numerical methods. Well known numerical methods such as Euler and Runge-Kutta sometimes fail because they generate oscillations, bifurcations, chaos and false steady states (see [8,13]). As a result we need to construct a more reliable solver that does not suffer through such drawbacks. ...
In this paper, a mathematical model of Visceral Leishmaniasis is considered. The model incorporates three populations, the human, the reservoir and the vector host populations. A detailed analysis of the model is presented. This analysis reveals that the model undergoes a backward bifurcation when the associated reproduction threshold is less than unity. For the case where the death rate due to VL is negligible, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. Noticing that the governing model is a system of highly nonlinear differential equations, its analytical solution is hard to obtain. To this end, a special class of numerical methods, known as the nonstandard finite difference (NSFD) method is introduced. Then a rigorous theoretical analysis of the proposed numerical method is carried out. We showed that this method is unconditionally stable. The results obtained by NSFD are compared with other well-known standard numerical methods such as forward Euler method and the fourth-order Runge–Kutta method. Furthermore, the NSFD preserves the positivity of the solutions and is more efficient than the standard numerical methods.
... In this paper, we investigated the threshold dynamics of an SEAR alcoholism model with the form of continuous equations and discrete equations, noting that the discrete model is formulated by applying the Michens nonstandard finite difference scheme to the corresponding continuous model. With this method of discretization, as demonstrated by the theoretical and numerical results obtained in this paper, the dynamical properties of the continuous system can be preserved with much high efficiency when compared with the traditional schemes such as forward Euler and Runge-Kutta, which sometimes fail generating oscillations, bifurcations, and chaos, and even result in false steady states [35,36]. Specifically, not only the expressions of equilibria and the basic reproductive number but also the threshold dynamics are identical in both models (1) and (20), which confirm its consistency of the discrete model with the original continuous model in the qualitative properties. ...
In this paper, an alcoholism model of SEAR type with different susceptibilities due to public health education is investigated, with the form of continuous differential equations as well as discrete differential equations by applying the Mickens nonstandard finite difference (NSFD) scheme to the continuous equations. Threshold dynamics of the continuous model are performed by constructing Lyapunov functions. The analysis of a discrete model indicates that the alcohol-free equilibrium is globally asymptotically stable if the basic reproductive number R0<1 , and conversely, the alcohol-present equilibrium is globally asymptotically stable if R0>1 , revealing the consistency and efficiency of the discrete model to preserve the dynamical properties of the corresponding continuous model. In addition, stability preserving and the impact of the parameters related with public health education are conducted by numerical simulations.
... Рис. 2. Хаотическая цепь с индуктивным элементом на основе системы Спротта Case D что говорит о корректности разработанной электрической схемы. Небольшая разница объясняется влиянием SPICEмоделей примененных радиодеталей и особенностями численного интегрирования: так, в Multisim используется многошаговый метод Гира 6-го порядка[11]. ...
The purpose of the work consists in the investigation of the possibility to use a random oscillator with the inductive coil as an inductive sensor. The work problems: methods development for the synthesis of a random oscillator with the inductive coil; development and investigation of methods for the analysis of oscillations in a random circuit at target bringing to a sensitive coil; an experimental confirmation of the results obtained theoretically. The investigation methods used in the work: a simulation, a computer experiment, a nature experiment. The investigation results: a confirmation of the hypothesis, lying in the basis of the investigation, on the applicability of random oscillators in sensor applications; a procedure for the synthesis of random oscillators with the inductive coil based on a specified random system of differential equations; data on the applicability of different kinds of the analysis of oscillations for the emphasis of information on the parameter measured, methods for these data processing. The novelty of this paper consists in that this is the first Russian language work on random sensors – a new promising direction of investigations in the field on non-linear dynamics. In this work for the first time there is used a random oscillator for the prototype formation of an approximation industrial sensor with a cup core. Conclusions: 1. On the basis of a random oscillator specified with a differential equation there may by synthesized a sensitive electric analogue circuit. 2. Using a simple geometrical analysis of an attractor it is possible to define a value of the parameter (inductivity) under measurement with high accuracy and to transform this value into data on a distance to an aim.
... The influence of numerical methods on discrete models of chaotic systems is widely studied. While highly accurate numerical methods for chaotic problems integration have been recently developed [1,2], some studies reveal the negative aspects of popular discretization techniques [3,4] and discover the additional properties introduced by numerical errors [5]. Thus, when new class of integration methods appears, the collateral numerical effects are of certain interest. ...
In this paper, we consider nonlinear integration techniques, based on direct Padé approximation of the differential equation solution, and their application to conservative chaotic initial value problems. The properties of discrete maps obtained by nonlinear integration are studied, including phase space volume dynamics, bifurcation diagrams, spectral entropy, and the Lyapunov spectrum. We also plot 2D dynamical maps to enlighten the features introduced by nonlinear integration techniques. The comparative study of classical integration methods and Padé approximation methods is given. It is shown that nonlinear integration techniques significantly change the behavior of discrete models of nonlinear systems, increasing the values of Lyapunov exponents and spectral entropy. This property reduces the applicability of numerical methods based on Padé approximation to the chaotic system simulation but it is still useful for construction of pseudo-random number generators that are resistive to chaos degradation or discrete maps with highly nonlinear properties.
... This algorithm caused a really abrupt change in the results, causing a periodic output. There are many related works that study numerical methods to make a qualitative difference in the results, as [14,15]. ...
The Chua's circuit has been considered as one of the most important paradigms for nonlinear science studies. Its simulations is usually undertaken by means of numerical methods under the rules of IEEE 754-2008 floating-point arithmetic standard. Although, it is well known the propagation error issue, less attention has been given to its consequences on the simulation of Chua's circuit. In this paper we presented a simulation technique for the Chua's circuit, it exhibits qualitative differences in traditional approaches such as RK3, RK4 and RK5. By means of the positive largest Lyapunov exponent we show that for the same initial condition and same set of parameters, we produce a periodical and a chaotic solution.
... This means that even though the solution will decay, it must be the solution of a problem that is more than 100% different to the original problem. Likewise, a substantial portion of the right-half plane will have |δ | > 1 for implicit Euler; this is why such numerical methods can suppress (actual) chaos [4]. In contrast, the implicit midpoint rule (with θ = 1/2) doesn't have such large 100% error zones (although it does, near the portions of the real axis where |µ| > 2). ...
We show how to compute the \emph{optimal relative backward error} for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of a general approach that uses results from optimal control theory to compute optimal residuals, but elementary methods can also be used here because the problem is so simple. This analysis produces new insight into the numerical solution of stiff problems.
... He has pioneered to notice that chaos could be an artefact of finite precision in digital computer. On the other hand, Corless et al. [4] have demonstrated the opposite effect. They have shown that some numerical methods may produce discrete dynamical systems that are not chaotic, even when the original continuous dynamical system is believed to be chaotic. ...
Interval arithmetic applied to simulation of dynamical systems has attracted a great deal of interest in recent years. Much of this research has been carried out in the calculation of fixed points or low-period windows for nonlinear discrete maps. This study proposes a novel interval computation based on a piecewise method to calculate periodic orbits for the logistic map. Using the cobweb plot, three rounding situations have been applied to a correct outward rounding, as required by interval arithmetic. The proposed method is compared with results in the literature and with the results obtained by means of the Mat-lab toolbox Intlab. The comparison is accomplished for nine case studies using the logistic map. Numerical results explicitly indicate that the proposed method produces intervals that are substantially narrower than those obtained with the traditional techniques.
... The connection between chaos and finite precision also deserves a remark here. Works such as in [28,[54][55][56][57][58][59][60][61][62][63][64] have been extensively investigated the effects of finite precision in the simulation of chaotic systems in many perspectives and in many sorts of systems. Using the LLE as a sort of propagation error measurement, we also estimate the maximum number of iterations wherein the simulation relies on a required precision or number of significant digits. ...
It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.
... It may also happen that the decrease in the time step may yield worse computational results. For instance, Corless et al. [15] pointed out that a true/correct chaotic orbit can be even suppressed. ...
In this part of the paper, the theory of nonlinear dynamics of flexible Euler-Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition allows for further generalization of the problem. The concept has been extended to a so-called noise-induced transition, which is a novel transition type exhibited by nonequilibrium systems embedded in a stochastic fluctuated medium, the properties of which depend on time and are influenced by external noise. Colored noise excitation of a structural system treated as a system with an infinite number of degrees of freedom has been studied.
... In such direction, it has been reported that a simulation of the Chua's circuit using the same set of parameters and initial conditions presents chaotic and periodical oscillation depending of the numerical method applied [41] . This kind of chaos suppression according numerical methods is not a new issue and it has been already studied in works such as [7] . Another fact that points out for the importance of the reliability [14,34] and reproducibility [40,43] of numerical simulation is the fact that there are theoretical models with no possible real experiment. ...
Abstract This paper presents a procedure to detect unreliable computer simulations of recursive functions. The proposed method calculates a lower bound error which is derived from two different pseudo-orbits based on interval extensions. The interval extensions are generated by taking into account the associative property of multiplication, which keeps the same error bound. We have tested our approach on the logistic map using many different programming languages and simulation packages, including Matlab, Scilab, Octave, Fortran and C. In all cases, the number of iterates is significantly lower than that considered reliable in the existing literature. We have also used the lower bound error on the logistic map and on the polynomial NARMAX for the Rössler equations to estimate the largest Lyapunov exponent, which determines the critical simulation time that guarantees the reliability of the simulation.
... Stability simply if for no other reason than computational control becomes a practical necessity. Just as false instabilities can be introduced by computation, so too can false stability [Corless et al., 1991]. That is models may well be over stabilized, killing off their natural variability on long timescales. ...
While famous theoretical work has been done historically on climate, no precise testable physical theory for climate has ever emerged. That is because, among other reasons, the definition of the objective is imprecise. The most common definition of climate as averaged weather, is more cliché than definition. Average over what? Average in what way? Is there a function relating resulting averages to each other, or do the averages satisfy differential equations? There is not one but many divergent approaches to defining climate in terms of averages, which seem to coexist without mutual competition. The three primary approaches employ time averages, field averages, and model solution ensemble averages, respectively. Each is problematic in its own way. While it is easy to produce an average, finding equations that can stand on their own in terms of averaged quantities only is not straightforward. But such equations are the ultimate aim of a search for a theory of climate, examining the questions of what averaging rule over what physical quantities help point to what an actual theory for climate ought to be like. This paper discusses averaging and closure in other fields, such as kinetic theory and turbulence, and how they are relevant to a theory of climate. It suggests how we might learn from them, while identifying how these issues need more exploration in terms of the climate problem.
... But they can produce nonchaotic discrete images of chaotic solutions of differential equations. This was shown for the backward Euler scheme on the RSssler system [5]. The stability theory of difference schemes is mostly a linear theory, i.e., the stability properties of difference schemes are mostly investigated on linear model problems ...
To understand the behavior of difference schemes on nonlinear differential equations, it seems desirable to extend the standard linear stability theory into a nonlinear theory. As a step in that direction, we investigate the stability properties of Euler-related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation.Among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other. We find that these schemes are superior to explicit schemes when they are stable and the blow-up time has not passed: for these λh-values they are dynamically faithful. When these schemes ‘turn unstable’, however, they have much less desirable properties than explicit or fully implicit schemes: they become simultaneously superstable and unstable. This is explained by the fact that these schemes are not self-adjoint: the linearly implicit self-adjoint scheme is dynamically faithful in an Euler-typical range of step sizes and gives correct stability for all step sizes.
... [15] Active research is under way in how computational schemes differ from the equations that they are meant to approximate. It is known that computational schemes cannot only produce error, but they can produce behaviors qualitatively different from the behaviors of the equations they are meant to emulate [e.g., Corless et al., 1991]. Modern study in this field examines the structural nature of the map and its underlying grid in the form of mathematical symmetries [e.g., Mansfield, 2006; Olver, 2000]. ...
1] This paper takes a novel approach to a known basic difficulty with computer simulations of nonlinear dynamical systems relevant to climate modeling. Specifically, we show by minimal examples how small systematic modeling errors might survive averaging over an ensemble of initial conditions. The resulting predictive errors can grow slowly enough initially that they may be overlooked without contradicting known behaviors on middle scales. However, they may nonetheless be significant on long timescales, given our current knowledge. Mathematical symmetry, which has been investigated for improving accuracy in computational algorithms, turns out to provide a novel perspective to this issue.
... Yee and Sweby (1995) have demonstrated that even for small time step sizes the numerical solutions obtained by various Euler, Runge-Kutta, predictor-corrector and trapezoidal schemes may produce spurious solutions which coexist with the correct solution having their own basins of attraction. Corless et al. (1991) demonstrated that the Euler numerical scheme can even suppress existing chaos. Investigation of steady state dynamics of possible chaotic systems requires large finite time. ...
Spectral element methods are high order accurate methods which have been successfully utilized for solving ordinary and partial differential equations. In this paper the space-time spectral element (STSE) method is employed to solve a simply supported modified Euler-Bernoulli nonlinear beam undergoing forced lateral vibrations. This system was chosen for analysis due to the availability of a reference solution of the form of a forced Duffing's equation. Two formulations were examined: i) a generalized Galerkin method with Hermitian polynomials as interpolants both in spatial and temporal discretization (HHSE), ii) a mixed discontinuous Galerkin formulation with Hermitian cubic polynomials as interpolants for spatial discretization and Lagrangian spectral polynomials as interpolants for temporal discretization (HLSE). The first method revealed severe stability problems while the second method exhibited unconditional stability and was selected for detailed analysis. The spatialh-convergence rate of the HLSE method is of order α=p
s+1 (wherep
s is the spatial polynomial order). Temporalp-convergence of the HLSE method is exponential and theh-convergence rate based on the end points (the points corresponding to the final time of each element) is of order 2p
T−1 ≤α≤2p
T+1 (wherep
T is the temporal polynomial order). Due to the high accuracy of the HLSE method, good results were achieved for the cases considered using a relatively large spatial grid size (4 elements for first mode solutions) and a large integration time step (1/4 of the system period for first mode solutions, withp
T=3). All the first mode solution features were detected including the onset of the first period doubling bifurcation, the onset of chaos and the return to periodic motion.
Two examples of second mode excitation produced homogeneous second mode and coupled first and second mode periodic solutions.
Consequently, the STSE method is shown to be an accurate numerical method for simulation of nonlinear spatio-temporal dynamical systems exhibiting chaotic response.
Computational chaos reports the artificial generation or suppression of chaotic behaviour in digital computers. There is a significant interest of the scientific community in analysing and understanding computational chaos of discrete and continuous systems. Notwithstanding, computational chaos in complex networks has received much less attention. In this article, we report computational chaos in a network of coupled logistic maps. We consider two types of networks, namely the Erdös–Rényi random network and the Barabási–Albert scale-free network. We show that there is an emergence of computational chaos when two different natural interval extensions are used in the simulation. More surprisingly, we also show that this chaos can be suppressed by an average of natural interval extensions, which can thus be considered as a filter to reduce the uncertainty stemming from the inherent finite precision of computer simulations.
The tendency of certain numerical discretization methods towards super-stability or instability is investigated in this work. To this end, the coupled ordinary differential equations which govern the kinetic behavior of a class of Advanced Heavy Water nuclear Reactors (AHWR) are taken into account. Single step explicit and implicit methods and the multi-step central difference scheme, also known as the leapfrog or the midpoint method, have been applied to these coupled ODEs. Pertaining stability regions have been delineated in the parametric plane characterized by the pair of inherent thermal feedback coefficients present in the model. To this effect, the bifurcation framework has been employed and the discretized schemes were compared to the original (continuous) system. Results pointed out that the implicit schemes enjoy a super-stable behavior for large enough time-steps while characterizing a broader stability region. The explicit scheme on the other hand is quite susceptible to an unstable dynamic regime. Besides, the super-stable behavior is quite more pronounced in the midpoint rule owing basically to the higher order accuracy thereof.
Composition algorithms make up a prospective class of methods for solving ordinary differential equations. Their main advantage is an ability to retain some properties of the simulated continuous systems, e.g. phase space volume. Meanwhile, computational costs of composition solvers for non-Hamiltonian systems are high because the implicit midpoint rule should be used as a basic method. This also complicates the development of embedded applications based on the numerical solution of ODEs, such as hardware chaos generators. In this article, a new semi-explicit composition methods are proposed. The stability regions for different composition algorithms were plotted and a memcapacitor circuit was studied as a test problem. Computational experiments reveal the superior properties of semi-explicit composition algorithms as a hardware-targeted ODE solvers. The obtained results imply that the development of semi-explicit composition algorithms is a step towards construction a new generation of simulation software for nonlinear dynamical systems and embedded chaos generators.
Composition algorithms make up a prospective class of methods for solving ordinary differential equations. Their main advantage is an ability to retain some properties of the simulated continuous systems, e.g. phase space volume. Meanwhile, computational costs of composition solvers for non-Hamiltonian systems are high because the implicit midpoint rule should be used as a basic method. This also complicates the development of embedded applications based on the numerical solution of ODEs, such as hardware chaos generators. In this paper, a new semi-explicit composition methods are proposed. The stability regions for different composition algorithms were plotted and a memcapacitor circuit was studied as a test problem. Computational experiments reveal the superior properties of semi-explicit composition algorithms as a hardware-targeted ODE solvers. The obtained results imply that the development of semi-explicit composition algorithms is a step towards construction a new generation of simulation software for nonlinear dynamical systems and embedded chaos generators.
The interdisciplinary journal publishes original and new results on recent developments, discoveries and progresses on Discontinuity, Nonlinearity and Complexity in physical and social sciences. The aim of the journal is to stimulate more research interest for exploration of discontinuity, complexity, nonlinearity and chaos in complex systems. The manuscripts in dynamical systems with nonlinearity and chaos are solicited, which includes mathematical theories and methods, physical principles and laws, and computational techniques. The journal provides a place to researchers for the rapid exchange of ideas and techniques in discontinuity, complexity, nonlinearity and chaos in physical and social sciences.
No length limitations for contributions are set, but only concisely written manuscripts are published. Brief papers are published on the basis of Technical Notes. Discussions of previous published papers are welcome.
Topics of Interest
Complex and hybrid dynamical systems
Discontinuous dynamical systems (i.e., impulsive, time-delay, flow barriers)
Nonlinear discrete systems and symbolic dynamics
Fractional dynamical systems and control
Stochastic dynamical systems and randomness
Complexity, self-similarity and synchronization in nonlinear physics
Nonlinear phenomena and physical mechanisms
Stability, bifurcation and chaos in complex systems
Hydrodynamics, turbulence and complexity mechanism
Nonlinear waves and soliton
Dynamical networks
Combinatorial aspects of dynamical systems
Biological dynamics and biophysics
Pattern formation, social science and complexity
By combining chaotic system and DNA sequence operations, an image cryptosystem is proposed. To generate sequences with better chaotic characteristics, a new spatiotemporal chaotic system is constructed by employing the Logistic-Sine system (LSS) in the coupled map lattice (CML). In the cryptosystem, the original image is firstly diffused through exclusive or with a key image transformed from the constructed spatiotemporal chaotic sequences. Furthermore, DNA deletion and DNA insertion pseudo-operations are used to confuse the DNA-encoded diffused image under the control of the key streams. The encrypted image is obtained after decoding the confused DNA image. Experimental results and performance analysis demonstrate that the proposed image cryptosystem has acceptable speed, good robustness and outperforms some existing image encryption schemes to counteract the recognized attacks.
In the present study, the susceptibility of the forward and the backward Euler methods to computational chaos and superstability is investigated via the means of both a theoretical analysis and numerical experiments. A linear stability analysis of the fixed points and the periodic orbits of the maps induced by these methods asserts that, for large enough time-steps Δt, these maps undergo bifurcations and as result the acquired solutions are spurious. More specifically, it is shown that the backward Euler method suppresses chaotic behavior, whereas the forward Euler renders all linearly stable fixed points and periodic orbits of its induced map linearly unstable. Numerical experiments that illustrate the validity of the theoretical analysis are also presented and discussed. For the forward Euler method, in particular, the computation of bifurcation diagrams, the Maximum Lyapunov exponent and the Kolmogorov–Sinai entropy suggest that it can engender computational chaos.
Discrete schemes, used to perform time integration of ODE’s, are expected to exhibit qualitatively ‘true’ dynamics in terms of the solutions and their stability. In past years, it has been discovered that such discretizations may cause spurious steady states and some explicit schemes may produce ‘computational chaos.’ In this study, we show that implicit time integration schemes, such as the backward Euler method, can also produce computationally chaotic solutions. Furthermore, we show that the opposite phenomenon may also take place both for explicit and for implicit schemes: computationally generated ‘spurious order’ may replace the true chaotic solution before the scheme becomes linearly unstable. The numerical solution may become chaotic again as the discretization step is further increased. The spurious computational order and chaos are discussed by solving low-dimensional dynamical systems, as well as a large system of ODE representing the solution to the Navier-Stokes equation. Our results support the point of view that the deviations in the behavior of the computed solution from the true solution has deterministic character with the time step playing the role of an artificial bifurcation parameter.
[1] Despite best efforts, there still is no physical theory for climate based on the physics of the laboratory regime (i.e., fluid mechanics, radiative transfer, classical thermodynamics, etc.). This paper builds from previous discussions on how laboratory-regime assumptions may lock our current theoretical efforts into the laboratory regime and how we might get around this problem. Using ultralong time photographic exposures (known as solargraphs) for inspiration, it draws into question classical thinking about intensive thermodynamic variables for theoretical climate purposes while using the fact that physical flows remain well defined even for climate regimes. These flows are characterized here as “generalized wind.” A simple example based on radiative energy and entropy transfer illustrates how these generalized wind fields can partially replace what is lost in moving away from laboratory-regime physics. These winds are shown to carry the dynamics in a modified form of radiation-like fluid dynamics that, together with radiation, might be possible to close in climate regimes.
Numerical schemes based on first-order methods and a second-order method are developed and analysed for solving the non-linear Froude's Pendulum system. The second-order method is developed by taking a linear combination of three first-order methods. Though implicit in nature, with the resulting improvements in stability, the methods are applied explicitly. These methods are extended to a more general second-order ordinary differential equation.Stability of the numerical methods is approached from the point of view of dynamical systems. It is found that the methods are stable under the same conditions as those required by the linearized schemes in the neighbourhood of constant, stable, fixed points of the underlying initial-value problem. For the second-order method, this result is found to hold for any chosen value of the step-length.A major advantage of using the second-order method is the absence of solutions to the discrete finite-difference equation that do not correspond to the continuous dynamical system. This contrasts starkly with the Runge-Kutta methods which are found to produce solutions that converge to a false asymptote or wrong solutions in a deceptively smooth manner if the step-length is not rightly chosen.A selection of examples are carried out on the non-linear Froude's pendulum system to illustrate the differing predictions obtained by the methods.
This paper is devoted to further exploration of the PSL technique developed earlier in its companion paper. To start with, it is shown that the method is not only applicable for obtaining one-periodic orbits, but can be made use of for obtaining every conceivable orbit, including subharmonic or periodic-doubled orbits, quasi-periodic orbits and even chaotic orbits. This is numerically illustrated by constructing various orbits of Ueda's, Duffing–Holmes' and van der Pol's oscillators. Next, the separatrix, that separates the basins of attraction of the stable limit cycles of Duffing-Holmes' oscillator, is constructed using the PSL procedure. A possibility of predicting the chaotic diffusion of trajectories based on a heuristic argument of a near-tangency of stable and unstable limit cycles of Duffing-Holmes' oscillator is also discussed. Finally, the PSL scheme is made use of to compute various characteristic quantities such as Fourier spectra, Liapunov characteristic exponents and probability density functions. Many new results are presented to establish the versatility of the PSL method.
Four-wave mixing equations in photorefractive media are approximated by different dynamical models and treated by different numerical methods. It is shown that the onset of instabilities and irregular behaviour in the same crystal, with a single wave mixing region, may be dependent both on the model used and the numerical method applied. Long-time irregular dynamics following from any finite-order difference schemes should be viewed with caution.
Interval mathematics is applied to computational problems in dynamical systems theory. We present methods of global analysis that automatically verify the existence and uniqueness of all periodic orbits of fixed period of iterated mappings. Furthermore, the method is generalised to include parameter space in the global analysis. A method is developed that computes a guaranteed enclosure of invariant sets of iterated mappings; the invariant sets may contain chaotic attractors or chaotic repellers. The methods are applied to the logistic map, the Hénon map and a two-dimensional trigonometric map.
A prototype equation to the Lorenz model of turbulence contains just one (second-order) nonlinearity in one variable. The flow in state space allows for a “folded” Poincaré map (horseshoe map). Many more natural and artificial systems are governed by this type of equation.
Stability of numerical methods for nonlinear autonomous ordinary differential equations is approached from the point of view
of dynamical systems. It is proved that multistep methods (with nonlinear algebraic equations exactly solved) with bounded
trajectories always produce correct asymptotic behaviour, but this is not the case with Runge-Kutta. Examples are given of
Runge-Kutta schemes converging to wrong solutions in a deceptively ‘smooth’ manner and a characterization of such two-stage
methods is presented. PE(CE)m schemes are examined as well, and it is demonstrated that they, like Runge-Kutta, may lead to false asymptotics.
This paper reports the use of the Gauss map from the theory of simple continued fractions as an example of a chaotic discrete dynamical system. Because of the simplicity of the map and the wealth of classical mathematical results, we are able to gain insight into the interaction between exact dynamical systems and their floating-point simulations. We calculate the correlation dimension and the capacity dimension of the Gauss map, and use these to examine current reconstruction techniques.
We consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set Λ of arbitrary shape. Under the assumption of uniform asymptotic stability of Λ in the sense of Lyapunov, we show that discretized versions of the dynamical system involving one-step numerical methods have nearby attracting sets Λ(h), which are also uniformly asymptotically stable. Our proof uses the properties of a Lyapunov function which characterizes the stability of Λ.
Numerical investigations of dynamical systems allow one to give estimates of the rate of divergence of nearby trajectories, by means of a quantity which is usually assumed to be related to the Kolmogorov (or metric) entropy. In this paper it is shown first, on the basis of mathematical results of Oseledec and Piesin, how such a relation can be made precise. Then, as an example, a numerical study of the Kolmogorov entropy for the Hénon-Heiles model is reported.
We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.
It is shown how the existence of low-dimensional chaotic dynamical systems describing turbulent fluid flow might be determined experimentally. Techniques are outlined for reconstructing phase-space pictures from the observation of a single coordinate of any dissipative dynamical system, and for determining the dimensionality of the system's attractor. These techniques are applied to a well-known simple three-dimensional chaotic dynamical system.
The discretisation of the ordinary nonlinear differential equation by the entral difference scheme is studied for fixed mesh size. In the usual numerical computation, this method produces some “ghost solution” for the long range calculation. Regarding this discretisation as a dynamical system in R2, these pathological behaviors are shown to be a kind of “chaos” in the dynamical system for any mesh size. Moreover, some combination of the central difference scheme and the Euler's scheme is studied for the above equation. It gives some motivation for Hénon's model. The usual discretisation of a second order differential equation are studied also. It gives some chaotic behaviors numerically which is similar to the behavior of the orbits of the system of differential equations proposed by Hénon-Heiles.
The existence of chaotic orbits is proved for area preserving diffeomorphisms of the plane, defined by polynomials of degree two, derived from the discretized logistic equation.
Chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment τ. In two simple examples we find that, as τ increases, chaos first sets in when attractor A acquires two distinct points that map to the same point. This happens when A acquires slopes of the same sign, in a rectifying coordinate system, at two consecutive intersections with the critical curve. Chaotic and quasi-periodic behavior may then alternate within a range of τ before computational instability finally prevails. Bifurcations to and from chaos and transitions to computational instability are highly scheme-dependent, even among differencing schemes of the same order. Systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings.
The chaotic attractor of a periodically forced Van der Pol oscillator (Shaw variant) is observed in digital simulation, and is made to vanish in a blue sky catastrophe by increasing a constant (bias) term in the force. The detailed bifurcation diagram, based on extensive simulations, reveals the involvement of the homoclinic outset of a nearby limit cycle of saddle type.
We show that a one-step method as applied to a dynamical system with a hyperbolic periodic orbit, exhibits an invariant closed curve for sufficiently small step size. This invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit. The dynamics of the one-step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression. Our results complement those of [2, 3] where one-step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a Hopf bifurcation.
We show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one-step methods or multi-step methods satisfying a strong root condition. This means that any continuous trajectory can be approximated by an appropriate discrete trajectory, and vice versa, to the correct order of convergence and uniformly on arbitrarily large time intervals. In particular, the stable and unstable manifolds of the discretization converge to their continuous counterparts.
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P.E. Kloeden and J. Lorenz, SIAM J. Numer. Anal. 23 changed from unstable to stable, and apparently the (1986) 986. stability of periodic solutions is similarly changed.
Numerical solution of boundary value problems for ordinary 5 We have shown here that, complementary to the
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U.M. Ascher, R.M.M. Mattheij and R.D. Russell, Numerical solution of boundary value problems for ordinary 5. Conclusions differential equations (Prentice-Hall, Englewood Cliffs, 1988). We have shown here that, complementary to the
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W.J. Beyn, SIAM J. Numer. Anal. 24 (1987) 1095. enough step sizes the stability of fixed points is
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Physica D 4 (1982) 407. in a continuous dynamical system and can introduce
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Georgia Institute ofTechnology, Tech. merical methods, that of spurious apparent chaos in-Rep. Math. 050290-039. troduced by roundoff error. This problem must be
- L Dieci
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For small enough step sizes
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The recovery of Lyapunov exponents from time series data
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