# Julien Clinton SprottUniversity of Wisconsin–Madison | UW · Department of Physics

Julien Clinton Sprott

PhD

## About

404

Publications

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Introduction

Additional affiliations

September 1970 - August 1973

September 1964 - present

Education

September 1964 - June 1969

September 1960 - June 1964

## Publications

Publications (404)

We revisit the laser model with cavity loss modulation, from which evidence of chaos and generalized multistability was discovered in 1982. Multistability refers to the coexistence of two or more attractors in nonlinear dynamical systems. Despite its relative simplicity, the adopted model shows us how the multistability depends on the dissipation o...

This issue is dedicated to the memory of Prof. Tenreiro Machado.
https://dergipark.org.tr/en/pub/chaos/issue/64884

Obtaining the master stability function is a well-known approach to study the synchronization in networks of chaotic oscillators. This method considers a normalized coupling parameter which allows for a separation of network topology and local dynamics of the nodes. The present study aims to understand how the dynamics of oscillators affect the mas...

Chaos Theory and Applications (March 2022 - Volume 4 - Issue 1)
https://dergipark.org.tr/en/pub/chaos/issue/63571
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1) Jun MA. "Chaos Theory and Applications：The Physical Evidence, Mechanism are Important in Chaotic Systems. "
2) Burak ARICIOĞLU, Sezgin KAÇAR. "Circuit Implementation and PRNG Applications of Time Delayed Lorenz System....

As a way to quantify the robustness of a chaotic system, a scheme is proposed to determine the extent to which the parameters of the system can be altered before the probability of destroying the chaos exceeds [Formula: see text]. The calculation uses a Monte-Carlo method and is applied to several common dissipative chaotic maps and flows with vary...

Based on the analysis of polarity balance and exhaustive computer searching, a series of symmetric chaotic flows is found for hosting conditional symmetry. Symmetric structure shapes the elegant symmetric phase trajectory, and conditional symmetry permits the convenience of embedding an extra set of coexisting symmetric attractors. Bifurcation anal...

The structure of a dynamical system is the key factor for investigating its multi-stability. Generally, a system can be divided into two categories; that is, a symmetric one and an asymmetric one. A dynamical system X˙=F(X)=(f1(X),f2(X),...,fN(X))(X=(x1,x2,...,xN)T) is symmetric if there exists a variable substitution: u1=-x1,u2=-x2,...,uk=-xk,ui=x...

As we discussed in the above chapters, many dynamical systems can produce similar attractors, specifically some of which [1–10] share the same Lyapunov exponents.

Exploring and establishing artificial neural networks
with electrophysiological characteristics and high computational
efficiency is a popular topic that has been explored for many years
in the fields of pattern recognition and computer vision. Inspired
by the working mechanism of the primary visual cortex, pulse�6 coupled neural networks (PCNNs) c...

Chaos Theory and Applications (November 2021 - Volume 3 - Issue 2)
https://dergipark.org.tr/en/pub/chaos/issue/58077

A simple memristive chaotic jerk system with one variable to represent the internal state is found. The proposed equilibria-free memristive system yields hidden chaotic oscillation in a narrow parameter space. A circuit is constructed that models the jerk system, and it shows agreement with the predicted oscillation. The new memristive jerk system...

Chaos Theory and Applications (June 2021 - Volume 3 - Issue 1)
https://dergipark.org.tr/tr/pub/chaos/issue/56378

Studying the growth of HIV virus in the human body as one of the fastest infectious viruses is very important. Using mathematical modeling can make experimental tests easier to process and evaluate. It can also help to predict the disease progress and provide a better insight into the virus development. In this study, a new nonlinear differential e...

Simulating and imitating the neuronal network of humans or mammals is a popular topic that has been explored for many years in the fields of pattern recognition and computer vision. Inspired by neuronal conduction characteristics in the primary visual cortex of cats, pulse-coupled neural networks (PCNNs) can exhibit synchronous oscillation behavior...

We study the synchronization of coupled identical circulant and non-circulant oscillators using single variable and different multi-variable coupling schemes. We use the master stability function to determine conditions for synchronization, in particular the necessary coupling parameter that ensures a stable synchronization manifold. We show that f...

Multi-stability is ubiquitous in dynamical systems, thereby attracting a great deal of interest for its potential threats or benefits to engineering systems and applications. A multi-stable system has much more complicated dynamics for its many-to-many mapping between system parameters and initial conditions.

Symmetric pairs of attractors are usually found in symmetric systems. However, it does not mean that all the coexisting behavior comes from symmetric structures. Compared with symmetric structure, asymmetric topology seems more common. Typically, for a dynamical system \(\dot{X}=F(X)=({{f}_{1}}(X),{{f}_{2}}(X),\cdots ,{{f}_{N}}(X))\) \((X={{({{x}_{...

For a hyperbolic system, generally the Šil’nikov criterion [1, 2, 3, 4, 5, 6] is applicable for proving the existence of chaos.

The many examples in the previous chapters should leave no doubt that hidden attractors are common in nonlinear dynamical systems. Remarkably, hidden attractors can have basins that fill the entire space with every initial condition on the attractor. Two such examples are shown here.

Critical slowing down is considered to be an important indicator for predicting critical transitions in dynamical systems. Researchers have used it prolifically in the fields of ecology, biology, sociology, and finance. When a system approaches a critical transition or a tipping point, it returns more slowly to its stable attractor under small pert...

Chaos Theory and Applications (November 2020-Volume 2-Issue 2)
https://dergipark.org.tr/tr/pub/chaos/issue/54264

By introducing an absolute value function for polarity balance, some new examples of chaotic systems with conditional symmetry are constructed that have hidden attractors. Coexisting oscillations along with bifurcations are investigated by numerical simulation and circuit implementation. Such new cases enrich the gallery of hidden chaotic attractor...

The memristor is a fundamental two-terminal electrical component unique in that it possesses the properties of non-linearity and memory, which are pervasive across natural systems. It has been proven to be in principle a viable substrate for novel dynamical systems showing chaotic behavior, but the recourse to abstract, idealized mathematical non-l...

One of the main applications for chaotic circuits is the production of aperiodic signals with many of the characteristics of noise for secure communications and similar uses. However, the probability distribution function (pdf) of such signals is usually far from Gaussian. This paper describes a new chaotic circuit based on the recently proposed si...

There are complex chaotic manifolds in practical nonlinear dynamical systems, especially in nonlinear circuits and chemical engineering. Any system attractor has its own geometric and physical properties, such as granularity, orientation, and spatiotemporal distribution. Polarity balance plays an important role in the solution of a dynamical system...

A half century after the discovery of the Lorenz attractor, people are still finding and publishing new chaotic systems with special properties. Surely we are nearing the time when additional examples are unnecessary... or maybe not. Like many others, I first learned about chaos in the 1980s after working several decades in a different field. I tho...

When the polarity reversal induced by offset boosting is considered, a new regime of a time-reversible chaotic system with conditional symmetry is found, and some new time-reversible systems are revealed based on multiple dimensional offset boosting. Numerical analysis shows that the system attractor and repellor have their own dynamics in respecti...

The Nosé–Hoover oscillator is a well-studied chaotic system originally proposed to model a harmonic oscillator in equilibrium with a heat bath at constant temperature. Although it is a simple three-dimensional system with five terms and two quadratic nonlinearities, it displays a rich variety of unusual dynamics, but it falls considerably short of...

A new three-dimensional chaotic flow is proposed in this paper. The system is the simplest chaotic flow that has a line of equilibria. The chaotic attractor of the system is very special with two slow and fast parts. In other words, the dynamic of the system is a combination of slow and fast states. The unique chaotic attractor of the system is inv...

This paper proposes a behavioral model for cells that shows their different dynamics from a high pluripotent stem cell to any distinct cell fate. The proposed model considers a cell as a black-box for a living system and tries to depict the presumed behaviors of the system. The model is a multistable iterated map with sensitive dependence on initia...

In this paper, some new three-dimensional chaotic systems are proposed. The special property of these autonomous systems is their identical eigenvalues. The systems are designed based on the general form of quadratic jerk systems with 10 terms, and some systems with stable equilibria. Using a systematic computer search, 12 simple chaotic systems wi...

This paper aims to investigate critical slowing down indicators in different situations where the system’s parameters change. Variation of the bifurcation parameter is important since it allows finding bifurcation points. A system’s parameters can vary through different functions. In this paper, five cases of bifurcation parameter variation are con...

In this paper, we propose a new model to describe variations in interpretation and perception of a simple sentence by different people. To show the understandability of a simple sentence in the prediction of future situations, the meaning of a sentence is modeled as a fuzzy if-then rule, and the fuzzy model is investigated in an iterative process....

In this paper, a new structure of chaotic systems is proposed. There are many examples of differential equations with analytic solutions. Chaotic systems cannot be studied with the classical methods. However, in this paper we show that a system that has a simple analytical solution can also have a strange attractor. The main goal of this paper is t...

In this note, we define four main categories of conservative flows: (a) those in which the dissipation is identically zero, (b) those in which the dissipation depends on the state of the system and is zero on average as a consequence of the orbits being bounded, (c) those in which the dissipation depends on the state of the system and is zero on av...

Symmetry is usually prevented by the broken balance in polarity. If the offset boosting returns the balance of polarity when some of the variables have their polarity reversed, the corresponding system becomes conditionally symmetric and in turn produces coexisting attractors with that type of symmetry. In this paper, offset boosting in one dimensi...

Fractals are intricate geometrical forms that contain miniature copies of themselves on ever smaller scales. This colorful book describes methods for producing an endless variety of fractal art using a computer program that searches through millions of equations looking for those few that can produce images having aesthetic appeal. Over a hundred e...

A symmetric pair of hyperchaotic attractors based on the 4-D Rössler system is constructed by adjusting the polarity information of some of its variables. By introducing a plane of equilibria into this system, an attractor and a repellor can be bridged. As a result, the proposed system is revised to be time-reversible, and one of the coexisting att...

Migraine is one of the primary headache disorders in a group of the ten most prevalent and disabling diseases. There are some valuable computational models of this disease which considered the onset and spatial patterns of migraine pain. Here we focus on dynamical transitions of this cyclic disease using the subnetworks which are essential in its c...

In this short communication, we comment on the recent report of a hidden attractor in the classical Lorenz system. We contend that the reported system gives instead a chaotic transient whose duration approaches infinity at a critical value of the parameters. We caution others who are searching for hidden attractors to consider carefully the possibi...

Classical indicators of tipping points have limitations when they are applied to an ecological and a biological model. For example, they cannot correctly predict tipping points during a period-doubling route to chaos. To counter this limitation, we here try to modify four well-known indicators of tipping points, namely the autocorrelation function,...

Two simple chaotic maps without equilibria are proposed in this paper. All nonlinearities are quadratic and the functions of right-hand side of the equations are continuous. The procedure of their design is explained and dynamical properties of them such as return map, bifurcation diagram, Lyapunov exponents, and basin of attraction are investigate...

Purpose:
The purpose of this study is to introduce a chaos level test to evaluate linear and nonlinear voice type classification method performances under varying signal chaos conditions without subjective impression.
Study design:
Voice signals were constructed with differing degrees of noise to model signal chaos. Within each noise power, 100...

In this paper, we modify the Sprott M chaotic system to provide infinitely many co-existing attractors by replacing the offset boosting parameter with a periodic function giving what we call a self-reproducing system. Consequently, a chaotic signal with either polarity can be obtained by selecting different initial conditions. Various periodic func...

By introducing trigonometric functions in a 4-D hyperchaotic snap system, infinite 1-D, 2-D, and 3-D lattices of hyperchaotic strange attractors were produced. Furthermore a general approach was developed for constructing self-reproducing systems, in which infinitely many attractors share the same Lyapunov exponents. In this case, cumbersome consta...

This work describes the simplest chaotic system with a hyperbolic sine non-linearity, accompanied by analysis of Lyapunov exponents, bifurcations, and stability. The corresponding simple chaotic circuit using only diodes and linear components is designed and implemented. Finally, an application of the system to spread spectrum communication based o...

Epilepsy is a long-term chronic neurological disorder that is characterized by seizures. One type of epilepsy is simple partial seizures that are localized to one area on one side of the brain, especially in the temporal lobe, but some may spread from there. GABA (gamma-aminobutyric acid) is an inhibitory neurotransmitter that is widely distributed...

For non-invasively investigating the interaction between insulin and glucose, mathematical modeling is very helpful. In this paper, we propose a new model for insulin-glucose regulatory system based on the well-known prey and predator models. The results of previous researches demonstrate that chaos is a common feature in complex biological systems...

Although chaotic signals are considered to have great potential applications in radar and communication engineering, their broadband spectrum makes it difficult to design an applicable amplifier or an attenuator for amplitude conditioning. Moreover, the transformation between a unipolar signal and a bipolar signal is often required. In this paper,...

A new dynamical system based on Thomas' system is described with infinitely many strange attractors on a 3-D spatial lattice. The mechanism for this multistability is associated with the disturbed offset boosting of sinusoidal functions with different spatial periods. Therefore, the initial condition for offset boosting can trigger a bifurcation, a...

Dynamical systems with special properties are continually being proposed and studied. Many of these systems are variants of the simple harmonic oscillator with nonlinear damping. This paper characterizes these systems as a hierarchy of increasingly complicated equations with correspondingly interesting behavior, including coexisting attractors, cha...

A recent strand of macroeconomic literature has placed sentiment fluctuations at the forefront of the academic debate about the foundations of business cycles. Waves of optimism and pessimism influence the decisions of investors and consumers, and they might therefore be interpreted as a driving force for the performance of the economy in the short...

Periodic trigonometric functions are introduced in 2-D offset-boostable chaotic flows to generate an infinite 2-D lattice of strange attractors. These 2-D offset-boostable chaotic systems are constructed based on standard jerk flows and extended to more general systems by exhaustive computer searching. Two regimes of multistability with a lattice o...

Multistability exists in various regimes of dynamical systems and in different combinations, among which there is a special one generated by self-reproduction. In this paper, we describe a method for constructing self-reproducing systems from a unique class of variable-boostable systems whose coexisting attractors reside in the phase space along a...

Strange attractors have been extensively studied, but the same is not true for strange repellors. Some time-reversible systems have repellors that mirror their corresponding attractors and that exchange roles when time is reversed. In this paper, a conversion operator is introduced by which an easy transformation can be constructed between such a t...

A novel chaotic system is explored in which all terms are quadratic except for a linear function.
The slope of the linear function rescales the amplitude and frequency of the variables linearly
while its zero intercept allows offset boosting for one of the variables. Therefore, a free-controlled
chaotic oscillation can be obtained with any desired...

In this paper, we describe a periodically-forced oscillator with spatially-periodic damping. This system has an infinite number of coexisting nested attractors, including limit cycles, attracting tori, and strange attractors. We are aware of no similar example in the literature.

In this paper, we study a system of two Rossler oscillators coupled through a
time-varying link, periodically switching between two values. We analyze the
system behavior with respect to the frequency of the switching. By applying an
averaging technique under the hypothesis of a high switching frequency, we find
that although each value of the coup...

Transitions from one dynamical regime to another one are observed in many complex systems, especially biological ones. It is possible that even a slight perturbation can cause such a transition. It is clear that this can happen to an object when it is close to a tipping point. There is a lot of interest in finding ways to recognize that a tipping p...

In the current study, a novel model for human memory is proposed based on the chaotic dynamics of artificial neural networks. This new model explains a biological fact about memory which is not yet explained by any other model: There are theories that the brain normally works in a chaotic mode, while during attention it shows ordered behavior. This...

Perpetual Points (PPs) have been introduced as an interesting new topic in nonlinear dynamics, and there is a conjecture that these points can be used to find hidden attractors. This note demonstrates some examples where PPs cannot locate their hidden attractors.

We report on the finding of hidden hyperchaos in a 5D extension to a known 3D self-exciting homopolar disc dynamo. The hidden hyperchaos is identified through three positive Lyapunov exponents under the condition that the proposed model has just two stable equilibrium states in certain regions of parameter space. The new 5D hyperchaotic self-exciti...

Perpetual points represent a new interesting topic in the literature of nonlinear dynamics. This paper introduces some chaotic flows with four different structural features from the viewpoint of fixed points and perpetual points.

In 1979, Moffatt pointed out that the conventional treatment of the simplest self-exciting homopolar disc dynamo has inconsistencies because of the neglect of induced azimuthal eddy currents, which can be resolved by introducing a segmented disc dynamo. Here we return to the simple dynamo system proposed by Moffatt, and demonstrate previously unkno...

Using a systematic computer search, four simple chaotic flows with cubic nonlinearities were found that have the unusual feature of having a curve of equilibria. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

Asymmetric dynamical systems sometimes admit a symmetric pair of coexisting attractors for reasons that are not readily apparent. This phenomenon is called conditional symmetry and deserves further explanation and exploration. In this paper, a general method for constructing such systems is proposed in which the asymmetric system restores its origi...

Symmetric pairs of coexisting attractors are commonly found in symmetric dynamical systems when symmetry breaking occurs. By contrast, asymmetric bistability is rarely reported in either symmetric or asymmetric dynamical systems because such behavior typically occurs in narrow regions of parameter space and thus is often unnoticed. This paper descr...

A crisis of amplitude control can occur when a system is multistable. This paper proposes a new chaotic system with a line of equilibria to demonstrate the threat to amplitude control from multistability. The new symmetric system has two coefficients for amplitude control, one of which is a partial amplitude controller, while the other is a total a...

Using a systematic computer search, twelve simple three-dimensional chaotic flows were found that have surfaces of equilibria. Although there are some four-dimensional systems with surfaces of equilibria, there is no such system in three-dimensional state space reported in the literature. Such systems are not difficult to design, but they can have...

We describe the application of adaptive (variable time step) integrators to stiff differential equations encountered in many applications. Linear harmonic oscillators subject to nonlinear thermal constraints can exhibit either stiff or smooth dynamics. Two closely related examples, Nosé's dynamics and Nosé–Hoover dynamics, are both based on Hamilto...

In literature many chaotic systems, based on third-order jerk equations with different nonlinear functions, are available. A jerk system is taken to be a part of dynamical systems that can exhibit regular and chaotic behavior. By extension, a hyperjerk system can be described as a dynamical system with nth-order ordinary differential equations wher...