Simple Chaotic Hyperjerk System

2016 ◽  
Vol 26 (11) ◽  
pp. 1650189 ◽  
Author(s):  
Fatma Yildirim Dalkiran ◽  
J. C. Sprott
Keyword(s):  
Dynamical System ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Nonlinear Function ◽  
Nonlinear Functions ◽  
Third Order ◽  
Experimental Realization ◽  
Numerical Studies ◽  
Hyperjerk System ◽  
Jerk System

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 [Formula: see text]th-order ordinary differential equations where [Formula: see text] is 4 or up to. Hyperjerk systems have been investigated in literature in the last decade. This paper consists of numerical studies and experimental realization on FPAA for fourth-order hyperjerk system with exponential nonlinear function.

IMPULSIVE SYNCHRONIZATION FOR A NEW CHAOTIC OSCILLATOR

2007 ◽  
Vol 17 (02) ◽  
pp. 617-623 ◽  
Author(s):  
SINUHÉ BENÍTEZ ◽  
LEONARDO ACHO
Keyword(s):  
Impulsive Control ◽  
Chaotic Behavior ◽  
Polynomial System ◽  
Van Der Pol Oscillator ◽  
Chaotic Oscillator ◽  
Third Order ◽  
Van Der Pol ◽  
Impulsive Synchronization ◽  
Jerk System ◽  
Fourier Spectrum Analysis

Synchronization for a new proposed chaotic system based on impulsive control theory is presented. This new chaotic oscillator is a third order polynomial system (Jerk system), which was developed after the addition of a third state and innovation terms to the well known second order Van der Pol oscillator. The chaotic behavior of this new system is confirmed by using Lyapunov exponents, Poincaré maps, Fourier spectrum analysis and numerical experiments. Impulsive synchronization is achieved using just one channel of communication.

scholarly journals ON A DYNAMICAL SYSTEM WITH MULTIPLE CHAOTIC ATTRACTORS

2007 ◽  
Vol 17 (09) ◽  
pp. 3235-3251 ◽  
Author(s):  
XIAODONG LUO ◽  
MICHAEL SMALL ◽  
MARIUS-F. DANCA ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Monte Carlo ◽  
Chaotic Behavior ◽  
Hypothesis Test ◽  
Surrogate Data ◽  
Chaotic Attractors ◽  
System A ◽  
Numerical Studies

The chaotic behavior of the Rabinovich–Fabrikant system, a model with multiple topologically different chaotic attractors, is analyzed. Because of the complexity of this system, analytical and numerical studies of the system are very difficult tasks. Following the investigation of this system carried out in [Danca & Chen, 2004], this paper verifies the presence of multiple chaotic attractors in the system. Moreover, the Monte Carlo hypothesis test (or, equivalently, surrogate data test) is applied to the system for the detection of chaos.

Identification of a Nonlinear Function in a Dynamical System

1993 ◽  
Vol 115 (4) ◽  
pp. 587-591 ◽  
Author(s):  
W. Messner ◽  
R. Horowitz
Keyword(s):  
Dynamical System ◽  
Integral Equation ◽  
Dynamical Systems ◽  
Mild Condition ◽  
Influence Function ◽  
Nonlinear Function ◽  
Disk Drive ◽  
Adaptive Method ◽  
Nonlinear Functions ◽  
Identification Technique

Using an adaptive method introduced in (Messner et al., 1991), a standard identification technique for linear systems can be extended to identify nonlinear functions in dynamical systems under a mild condition. Specifically, the assumption is that the nonlinear function can be represented as a integral equation of the first kind. The method identifies the nonlinear function indirectly by estimating the influence function of the integral equation. By analogy to linear methods the kernel of the integral equation serves as the “regressor,” while the influence function is the “parameter” to be identified. This paper focuses on an application to disk drive servos.

scholarly journals Experimental Realization of a Multiscroll Chaotic Oscillator with Optimal Maximum Lyapunov Exponent

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Esteban Tlelo-Cuautle ◽  
Ana Dalia Pano-Azucena ◽  
Victor Hugo Carbajal-Gomez ◽  
Mauro Sanchez-Sanchez
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Mathematical Description ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Quantitative Measure ◽  
Operational Amplifiers ◽  
Chaotic Oscillator ◽  
Maximum Lyapunov Exponent ◽  
Experimental Realization

Nowadays, different kinds of experimental realizations of chaotic oscillators have been already presented in the literature. However, those realizations do not consider the value of the maximum Lyapunov exponent, which gives a quantitative measure of the grade of unpredictability of chaotic systems. That way, this paper shows the experimental realization of an optimized multiscroll chaotic oscillator based on saturated function series. First, from the mathematical description having four coefficients (a, b, c, d1), an optimization evolutionary algorithm varies them to maximize the value of the positive Lyapunov exponent. Second, a realization of those optimized coefficients using operational amplifiers is given. Hereina, b, c, d1are implemented with precision potentiometers to tune up to four decimals of the coefficients having the range between 0.0001 and 1.0000. Finally, experimental results of the phase-space portraits for generating from 2 to 10 scrolls are listed to show that their associated value for the optimal maximum Lyapunov exponent increases by increasing the number of scrolls, thus guaranteeing a more complex chaotic behavior.

Simple Chaotic Flow with Circle and Square Equilibrium

2016 ◽  
Vol 26 (08) ◽  
pp. 1650137 ◽  
Author(s):  
Tomas Gotthans ◽  
Julien Clinton Sprott ◽  
Jiri Petrzela
Keyword(s):  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Nonlinear Differential Equations ◽  
Electrical Circuit ◽  
Hidden Attractors ◽  
Third Order ◽  
Chaotic Flow ◽  
New Class ◽  
Simple Systems ◽  
The Mathematical Model

Simple systems of third-order autonomous nonlinear differential equations can exhibit chaotic behavior. In this paper, we present a new class of chaotic flow with a square-shaped equilibrium. This unique property has apparently not yet been described. Such a system belongs to a newly introduced category of chaotic systems with hidden attractors that are interesting and important in engineering applications. The mathematical model is accompanied by an electrical circuit implementation, demonstrating structural stability of the strange attractor. The circuit is simulated with PSpice, constructed, and analyzed (measured).

LORENZ EQUATION AND CHUA’S EQUATION

1996 ◽  
Vol 06 (12b) ◽  
pp. 2443-2489 ◽  
Author(s):  
LADISLAV PIVKA ◽  
CHAI WAH WU ◽  
ANSHAN HUANG
Keyword(s):  
Lorenz System ◽  
Chaotic Behavior ◽  
Nonlinear Function ◽  
Mathematical Structure ◽  
Nonlinear Functions ◽  
Lorenz Equation ◽  
Functions Of Two Variables ◽  
The Lorenz System ◽  
Theoretical Results ◽  
Scalar Nonlinearity

The dynamical properties of two classical paradigms for chaotic behavior are reviewed—the Lorenz and Chua’s Equations—on a comparative basis. In terms of the mathematical structure, the Lorenz Equation is more complicated than Chua’s Equation because it requires two nonlinear functions of two variables, whereas Chua’s Equation requires only one nonlinear function of one variable. It is shown that most standard routes to cbaos and dynamical phenomena previously observed from the Lorenz Equation can be produced in Chua’s system with a cubic nonlinearity. In addition, we show other phenomena from Chua’s system which are not observed in the Lorenz system so far. Some differences in the topological geometric models are also reviewed. We present some theoretical results regarding Chua’s system which are absent for the Lorenz system. For example, it is known that Chua’s system is topologically conjugate to the class of systems with a scalar nonlinearity (except for a measure zero set) and is therefore canonical in this sense. We conclude with some reasons why Chua’s system can be considered superior or more suitable than the Lorenz system for various applications and studies.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

Numerical Studies on Trapped-Vortex Concepts for Stable Combustion

1998 ◽  
Vol 120 (1) ◽  
pp. 60-68 ◽  
Author(s):  
V. R. Katta ◽  
W. M. Roquemore
Keyword(s):  
Drag Coefficient ◽  
Reacting Flow ◽  
Flow Conditions ◽  
Third Order ◽  
Cold Flow ◽  
Flow Simulations ◽  
Numerical Studies ◽  
Dynamic Flows ◽  
Experimental Findings ◽  
Trapped Vortex

Spatially locked vortices in the cavities of a combustor aid in stabilizing the flames. On the other hand, these stationary vortices also restrict the entrainment of the main air into the cavity. For obtaining good performance characteristics in a trapped-vortex combustor, a sufficient amount of fuel and air must be injected directly into the cavity. This paper describes a numerical investigation performed to understand better the entrainment and residence-time characteristics of cavity flows for different cavity and spindle sizes. A third-order-accurate time-dependent Computational Fluid Dynamics with Chemistry (CFDC) code was used for simulating the dynamic flows associated with forebody-spindle-disk geometry. It was found from the nonreacting flow simulations that the drag coefficient decreases with cavity length and that an optimum size exists for achieving a minimum value. These observations support the earlier experimental findings of Little and Whipkey (1979). At the optimum disk location, the vortices inside the cavity and behind the disk are spatially locked. It was also found that for cavity sizes slightly larger than the optimum, even though the vortices are spatially locked, the drag coefficient increases significantly. Entrainment of the main flow was observed to be greater into the smaller-than-optimum cavities. The reacting-flow calculations indicate that the dynamic vortices developed inside the cavity with the injection of fuel and air do not shed, even though the cavity size was determined based on cold-flow conditions.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

On the Potential for Entangled States Between Chaotic Systems

2014 ◽  
Vol 24 (06) ◽  
pp. 1450077 ◽  
Author(s):  
Matthew A. Morena ◽  
Kevin M. Short
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Chaotic Systems ◽  
Entangled States ◽  
Future Research ◽  
Unstable Periodic Orbits ◽  
Qualitative Information ◽  
Chaotic Dynamical System ◽  
Naturally Occurring ◽  
External Intervention

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

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