scholarly journals Chaotic Dynamics by Some Quadratic Jerk Systems

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 227
Author(s):  
Mei Liu ◽  
Bo Sang ◽  
Ning Wang ◽  
Irfan Ahmad
Keyword(s):  
Hopf Bifurcation ◽  
Cross Sections ◽  
Chaotic Dynamics ◽  
Stable Equilibrium ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Chaotic Attractors ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
Subcritical Hopf Bifurcation

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

scholarly journals Scenario of the Birth of Hidden Attractors in the Chua Circuit

2017 ◽  
Vol 27 (12) ◽  
pp. 1730038 ◽  
Author(s):  
Nataliya V. Stankevich ◽  
Nikolay V. Kuznetsov ◽  
Gennady A. Leonov ◽  
Leon O. Chua
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic Dynamics ◽  
Equilibrium Points ◽  
Pitchfork Bifurcation ◽  
Dynamical Model ◽  
Chaotic Attractors ◽  
Hidden Attractors ◽  
Chua Circuit ◽  
Subcritical Hopf Bifurcation

Recently it was shown that in the dynamical model of Chua circuit both the classical self-excited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of self-excited and hidden attractors is studied. A pitchfork bifurcation is shown in which a pair of symmetric attractors coexist and merge into one symmetric attractor through an attractor-merging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed.

Hidden Hyperchaotic Attractors in a New 5D System Based on Chaotic System with Two Stable Node-Foci

2019 ◽  
Vol 29 (07) ◽  
pp. 1950092 ◽  
Author(s):  
Qigui Yang ◽  
Lingbing Yang ◽  
Bin Ou
Keyword(s):  
Chaotic System ◽  
Stable Equilibrium ◽  
Complex Dynamics ◽  
Dynamical Evolution ◽  
Stable Node ◽  
Hyperchaotic System ◽  
Hidden Attractors ◽  
Circuit Realization ◽  
Switching Algorithm ◽  
Linear Controllers

This paper reports some hidden hyperchaotic attractors and complex dynamics in a new five-dimensional (5D) system with only two nonlinear terms. The system is generated by adding two linear controllers to an unusual 3D autonomous quadratic chaotic system with two stable node-foci. In particular, the hyperchaotic system without equilibrium or with only one stable equilibrium can generate two kinds of hidden hyperchaotic attractors with three positive Lyapunov exponents. Numerical methods not only verify the existence of such attractors and hyperchaotic attractors, but also show the dynamical evolution of this system. The 5D system has self-excited attractors and two types of hidden attractors with the change of its parameter. The parameter switching algorithm is further utilized to numerically approximate the attractor. Specifically, the hidden hyperchaotic attractor can be approximated by switching between two self-excited chaotic attractors. Finally, the circuit realization results are consistent with the numerical results.

REGULAR AND CHAOTIC DYNAMICS OF THE LORENZ–STENFLO SYSTEM

2010 ◽  
Vol 20 (01) ◽  
pp. 145-152 ◽  
Author(s):  
JOÃO C. XAVIER ◽  
PAULO C. RECH
Keyword(s):  
Fixed Points ◽  
Gravity Waves ◽  
Chaotic Dynamics ◽  
System Parameter ◽  
Chaotic Attractors ◽  
Bifurcation Diagrams ◽  
Lorenz Equations ◽  
Precise Location ◽  
Acoustic Gravity Waves ◽  
The Stability

We analytically investigate the dynamics of the generalized Lorenz equations obtained by Stenflo for acoustic gravity waves. By using Descartes' Rule of Signs and Routh–Hurwitz Test, we decide on the stability of the fixed points of the Lorenz–Stenflo system, although without explicit solution of the eigenvalue equation. We determine the precise location where pitchfork and Hopf bifurcation of fixed points occur, as a function of the parameters of the system. Parameter-space plots, Lyapunov exponents, and bifurcation diagrams are used to numerically characterize periodic and chaotic attractors.

CHAOTIC DYNAMICS OF A FIVE-DIMENSIONAL NONLINEAR NETWORK

2007 ◽  
Vol 18 (03) ◽  
pp. 335-342
Author(s):  
XUEWEI JIANG ◽  
DI YUAN ◽  
YI XIAO
Keyword(s):  
Lyapunov Exponent ◽  
Power Spectrum ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Chinese Traditional Medicine ◽  
Bifurcation Diagrams ◽  
Nonlinear Network ◽  
Dynamical Behaviors ◽  
Lyapunov Exponent Spectrum ◽  
Period Doubling Bifurcations

The dynamics of a five-dimensional nonlinear network based on the theory of Chinese traditional medicine is studied by the Lyapunov exponent spectrum, Poincaré, power spectrum and bifurcation diagrams. The result shows that this system has complex dynamical behaviors, such as chaotic ones. It also shows that the system evolves into chaos through a series of period-doubling bifurcations.

Hopf Bifurcation and Hidden Attractors of a Delay-Coupled Duffing Oscillator

2015 ◽  
Vol 25 (12) ◽  
pp. 1550162 ◽  
Author(s):  
Huitao Zhao ◽  
Yiping Lin ◽  
Yunxian Dai
Keyword(s):  
Numerical Simulation ◽  
Hopf Bifurcation ◽  
Stable Equilibrium ◽  
Duffing Oscillator ◽  
Sufficient Conditions ◽  
Hidden Attractors ◽  
Characteristic Roots ◽  
Delayed System ◽  
Simulation Results ◽  
Spatio Temporal

In this paper, a delay-coupled Duffing equation is studied. By the characteristic roots technique, sufficient conditions are obtained for Hopf bifurcation occurrence. And the spatio-temporal patterns of the bifurcating periodic solutions are also obtained, some examples are given to demonstrate the theoretical analysis. Especially, the obtained numerical simulation results show that there are hidden attractors in this delayed system, which can coexist with stable equilibrium or stable bifurcating orbits.

BIFURCATION IN A PIECEWISE LINEAR CIRCUIT WITH SWITCHING BOUNDARIES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250034 ◽  
Author(s):  
ZHENGDI ZHANG ◽  
QINSHENG BI
Keyword(s):  
Periodic Orbits ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Generalized Jacobian ◽  
Linear Circuit ◽  
The Stability

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution.

scholarly journals EFFECT OF TIME DELAY IN A CANNIBALISTIC STAGE-STRUCTURED PREDATOR–PREY MODEL WITH HARVESTING OF AN ADULT PREDATOR: THE CASE OF LIONFISH

2019 ◽  
Vol 27 (04) ◽  
pp. 447-486
Author(s):  
BANAMALI MAJI ◽  
PANKAJ KUMAR TIWARI ◽  
SUDIP SAMANTA ◽  
SAMARES PAL ◽  
FRANCESCA BONA
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Chaotic Dynamics ◽  
Control Function ◽  
Period Doubling ◽  
Predation Rate ◽  
Predator Prey Model ◽  
Opportunistic Predator ◽  
The Stability ◽  
Coexistence Equilibrium

The progressive and increasing invasion of an opportunistic predator, the lionfish (Pterois volitans) has become a major threat for the delicate coral-reef ecosystem. The herbivore fish populations, in particular of Parrotfish, are taking the consequences of the lionfish invasion and then their control function on macro-algae growth is threatened. In this paper, we developed and analyzed a stage-structured mathematical model including P. volitans (lionfish), a cannibalistic predator, and a Parrotfish, its potential prey. As control upon the over predation, a rational harvest term has been considered. Further, to make the system more realistic, a delay in the growth rate of juvenile P. volitans population has been incorporated. We performed a global sensitivity analysis to identify important parameters of the system having significant correlations with the fishes. We observed that the system generates transcritical bifurcation, which takes the P. volitans-free equilibrium to the coexistence equilibrium on increasing the values of predation rate of adult P. volitans on Parrotfish. Further increase in the values of the predation rate of adult P. volitans on Parrotfish drives the system into Hopf bifurcation, which induces oscillation around the coexistence equilibrium. Moreover, the conversion efficiency due to cannibalism also has the property to alter the stability behavior of the system through Hopf bifurcation. The effect of time delay on the dynamics of the system is extensively studied and it is observed that the system develops chaotic dynamics through period-doubling oscillations for large values of time delay. However, if the system is already oscillatory, then the large values of time delay causes extinction of P. volitans from the system. To illustrate the occurrence of chaotic dynamics in the system, we drew the Poincaré map and also computed the Lyapunov exponents.

Analysis of a Chaotic Electrostatic Micro-Oscillator

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
S. Towfighian ◽  
G. R. Heppler ◽  
E. M. Abdel-Rahman
Keyword(s):  
Closed Loop ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Bifurcation Diagrams ◽  
Chaotic Oscillations ◽  
Frequency Sweep ◽  
Potential System ◽  
Loop Dynamics ◽  
The One

The closed-loop dynamics of a chaotic electrostatic microbeam actuator are presented. The actuator was found to be an asymmetric two-well potential system with two distinct chaotic attractors: one of which occurs predominantly in the lower well and a second that visits a lower-well orbit and a two-well orbit. Bifurcation diagrams obtained by sweeping the ac voltage amplitudes and frequency are presented. Period doubling, reverse period doubling, and the one-well chaos through period doubling are observed in amplitude sweep. In frequency sweep, period doubling, one-well, and two-well chaos, superharmonic resonances and on and off chaotic oscillations are found.

scholarly journals Lorenz- and Shilnikov-Shape Attractors in the Model of Two Coupled Parabola Maps

2021 ◽  
Vol 17 (2) ◽  
pp. 165-174
Author(s):  
E. Kuryzhov ◽  
◽  
E. Karatetskaia ◽  
D. Mints ◽  
◽  
...  
Keyword(s):  
Chaotic Dynamics ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
One Dimensional ◽  
Period Doubling Bifurcations

We consider the system of two coupled one-dimensional parabola maps. It is well known that the parabola map is the simplest map that can exhibit chaotic dynamics, chaos in this map appears through an infinite cascade of period-doubling bifurcations. For two coupled parabola maps we focus on studying attractors of two types: those which resemble the well-known discrete Lorenz-like attractors and those which are similar to the discrete Shilnikov attractors. We describe and illustrate the scenarios of occurrence of chaotic attractors of both types.

scholarly journals A Quadratic Fractional Map without Equilibria: Bifurcation, 0–1 Test, Complexity, Entropy, and Control

Electronics ◽  
2020 ◽  
Vol 9 (5) ◽  
pp. 748 ◽  
Author(s):  
Adel Ouannas ◽  
Amina-Aicha Khennaoui ◽  
Shaher Momani ◽  
Giuseppe Grassi ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Lyapunov Exponent ◽  
Test Phase ◽  
Research Topic ◽  
Chaotic Attractors ◽  
Maximum Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
Discrete Time Systems ◽  
And Control ◽  
Time Systems

Fractional calculus in discrete-time systems is a recent research topic. The fractional maps introduced in the literature often display chaotic attractors belonging to the class of “self-excited attractors”. The field of fractional map with “hidden attractors” is completely unexplored. Based on these considerations, this paper presents the first example of fractional map without equilibria showing a number of hidden attractors for different values of the fractional order. The presence of the chaotic hidden attractors is validated via the computation of bifurcation diagrams, maximum Lyapunov exponent, 0–1 test, phase diagrams, complexity, and entropy. Finally, an active controller with the aim for stabilizing the proposed fractional map is successfully designed.

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