scholarly journals A New Feigenbaum-Like Chaotic 3D System

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Huitao Zhao ◽  
Yiping Lin ◽  
Yunxian Dai
Keyword(s):  
Lyapunov Exponent ◽  
Autonomous System ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Period Doubling ◽  
Fractional Dimension ◽  
Largest Lyapunov Exponent ◽  
System A ◽  
Route To Chaos ◽  
Doubling Sequence

Based on Sprott N system, a new three-dimensional autonomous system is reported. It is demonstrated to be chaotic in the sense of having positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping, and period-doubling route to chaos are analyzed with careful numerical simulations. The obtained results also show that the period-doubling sequence of bifurcations leads to a Feigenbaum-like strange attractor.

Dynamic Analysis and Adaptive Sliding Mode Controller for a Chaotic Fractional Incommensurate Order Financial System

2017 ◽  
Vol 27 (13) ◽  
pp. 1750198 ◽  
Author(s):  
Ahmad Hajipour ◽  
Hamidreza Tavakoli
Keyword(s):  
Control Method ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Largest Lyapunov Exponent ◽  
Sliding Mode Controller ◽  
Adaptive Sliding Mode Controller ◽  
Uncertain Dynamics ◽  
Route To Chaos ◽  
Doubling Sequence

In this study, the dynamic behavior and chaos control of a chaotic fractional incommensurate-order financial system are investigated. Using well-known tools of nonlinear theory, i.e. Lyapunov exponents, phase diagrams and bifurcation diagrams, we observe some interesting phenomena, e.g. antimonotonicity, crisis phenomena and route to chaos through a period doubling sequence. Adopting largest Lyapunov exponent criteria, we find that the system yields chaos at the lowest order of [Formula: see text]. Next, in order to globally stabilize the chaotic fractional incommensurate order financial system with uncertain dynamics, an adaptive fractional sliding mode controller is designed. Numerical simulations are used to demonstrate the effectiveness of the proposed control method.

Study of Rehabilitation of Injured Knee Joint Applying Chaotic Theory in Human Body Motion

2007 ◽  
Vol 342-343 ◽  
pp. 581-584
Author(s):  
Byung Young Moon ◽  
Kwon Son ◽  
Jung Hong Park
Keyword(s):  
Time Series ◽  
Lyapunov Exponent ◽  
Angular Displacement ◽  
Three Dimensional ◽  
Gait Pattern ◽  
Body Motion ◽  
Largest Lyapunov Exponent ◽  
Chaos Analysis ◽  
Nonlinear Technique ◽  
Injured Knee

Gait analysis is essential to identify accurate cause and knee condition from patients who display abnormal walking. Traditional linear tools can, however, mask the true structure of motor variability, since biomechanical data from a few strides during the gait have limitation to understanding the system. Therefore, it is necessary to propose a more precise dynamic method. The chaos analysis, a nonlinear technique, focuses on understanding how variations in the gait pattern change over time. Healthy eight subjects walked on a treadmill for 100 seconds at 60 Hz. Three dimensional walking kinematic data were obtained using two cameras and KWON3D motion analyzer. The largest Lyapunov exponent from the measured knee angular displacement time series was calculated to quantify local stability. This study quantified the variability present in time series generated from gait parameter via chaos analysis. Gait pattern is found to be chaotic. The proposed Lyapunov exponent can be used in rehabilitation and diagnosis of recoverable patients.

Estimation of the Largest Lyapunov Exponent of Discontinuous Systems Using Chaos Synchronization

1999 ◽  
Author(s):  
Andrzej Stefanski ◽  
Jerzy Wojewoda ◽  
Tomasz Kapitaniak ◽  
John Brindley
Keyword(s):  
Lyapunov Exponent ◽  
Mechanical System ◽  
Dry Friction ◽  
Chaos Synchronization ◽  
Largest Lyapunov Exponent ◽  
Discontinuous Systems ◽  
System A ◽  
The Largest Lyapunov Exponent

Abstract Properties of chaos synchronization have been used for estimation of the largest Lyapunov exponent of a discontinuous mechanical system. A method for such estimation is proposed and an example is shown, based on coupling of two identical systems with dry friction which is modelled according to the Popp-Stelter formula.

scholarly journals Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Huayong Zhang ◽  
Shengnan Ma ◽  
Tousheng Huang ◽  
Xuebing Cong ◽  
Zichun Gao ◽  
...  
Keyword(s):  
Functional Response ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Period Doubling ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Routes To Chaos ◽  
Prey System ◽  
The One

We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.

scholarly journals Three-Dimensional Memristive Hindmarsh–Rose Neuron Model with Hidden Coexisting Asymmetric Behaviors

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Bocheng Bao ◽  
Aihuang Hu ◽  
Han Bao ◽  
Quan Xu ◽  
Mo Chen ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Electromagnetic Induction ◽  
Neuron Model ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Neuronal System ◽  
System A ◽  
Dynamical Behaviors ◽  
Electrical Activities

Since the electrical activities of neurons are closely related to complex electrophysiological environment in neuronal system, a novel three-dimensional memristive Hindmarsh–Rose (HR) neuron model is presented in this paper to describe complex dynamics of neuronal activities with electromagnetic induction. The proposed memristive HR neuron model has no equilibrium point but can show hidden dynamical behaviors of coexisting asymmetric attractors, which has not been reported in the previous references for the HR neuron model. Mathematical model based numerical simulations for hidden coexisting asymmetric attractors are performed by bifurcation analyses, phase portraits, attraction basins, and dynamical maps, which just demonstrate the occurrence of complex dynamical behaviors of electrical activities in neuron with electromagnetic induction. Additionally, circuit breadboard based experimental results well confirm the numerical simulations.

Coexistence of Multiple Attractors and Crisis Route to Chaos in a Novel Chaotic Jerk Circuit

2016 ◽  
Vol 26 (05) ◽  
pp. 1650081 ◽  
Author(s):  
J. Kengne ◽  
Z. T. Njitacke ◽  
A. Nguomkam Negou ◽  
M. Fouodji Tsostop ◽  
H. B. Fotsin
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Period Doubling ◽  
Chaotic Oscillations ◽  
Multiple Attractors ◽  
Coexistence Of Multiple Attractors ◽  
Theoretical Predictions ◽  
Route To Chaos ◽  
Model Equations ◽  
Good Agreement

In this paper, a novel autonomous RC chaotic jerk circuit is introduced and the corresponding dynamics is systematically investigated. The circuit consists of opamps, resistors, capacitors and a pair of semiconductor diodes connected in anti-parallel to synthesize the nonlinear component necessary for chaotic oscillations. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a linear transformation of model MO15 previously introduced in [Sprott, 2010]. The structure of the equilibrium points and the discrete symmetries of the model equations are discussed. The bifurcation analysis indicates that chaos arises via the usual paths of period-doubling and symmetry restoring crisis. One of the key contributions of this work is the finding of a region in the parameter space in which the proposed (“elegant”) jerk circuit exhibits the unusual and striking feature of multiple attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). Laboratory experimental results are in good agreement with the theoretical predictions.

scholarly journals Chaotic Attractor Generation via a Simple Linear Time-Varying System

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
Baiyu Ou ◽  
Desheng Liu
Keyword(s):  
Lyapunov Exponent ◽  
Chaotic Attractor ◽  
Linear Time ◽  
Three Dimensional ◽  
Underlying Mechanism ◽  
Largest Lyapunov Exponent ◽  
Time Varying ◽  
System Parameters ◽  
The Largest Lyapunov Exponent ◽  
Suitable System

A novel generation method of chaotic attractor is introduced in this paper. The underlying mechanism involves a simple three-dimensional time-varying system with simple time functions as control inputs. Moreover, it is demonstrated by simulation that various attractor patterns are generated conveniently by adjusting suitable system parameters. The largest Lyapunov exponent of the system has been obtained.

A Novel Cubic–Equilibrium Chaotic System with Coexisting Hidden Attractors: Analysis, and Circuit Implementation

2017 ◽  
Vol 27 (04) ◽  
pp. 1850066 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Christos Volos ◽  
Sajad Jafari ◽  
Tomasz Kapitaniak
Keyword(s):  
Chaotic System ◽  
Autonomous System ◽  
Chaotic Systems ◽  
Electronic Circuit ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
Hidden Attractors ◽  
Dynamical Properties

Chaotic systems with a curve of equilibria have attracted considerable interest in theoretical researches and engineering applications because they are categorized as systems with hidden attractors. In this paper, we introduce a new three-dimensional autonomous system with cubic equilibrium. Fundamental dynamical properties and complex dynamics of the system have been investigated. Of particular interest is the coexistence of chaotic attractors in the proposed system. Furthermore, we have designed and implemented an electronic circuit to verify the feasibility of such a system with cubic equilibrium.

A CHAOTIC SYSTEM WITH ONE SADDLE AND TWO STABLE NODE-FOCI

2008 ◽  
Vol 18 (05) ◽  
pp. 1393-1414 ◽  
Author(s):  
QIGUI YANG ◽  
GUANRONG CHEN
Keyword(s):  
Chaotic System ◽  
Chaotic Attractor ◽  
Autonomous System ◽  
Lorenz System ◽  
Complex Dynamics ◽  
Three Dimensional ◽  
Type Systems ◽  
Stable Node ◽  
Compound Structure ◽  
Chen System

This paper reports the finding of a chaotic system with one saddle and two stable node-foci in a simple three-dimensional (3D) autonomous system. The system connects the original Lorenz system and the original Chen system and represents a transition from one to the other. The algebraical form of the chaotic attractor is very similar to the Lorenz-type systems but they are different and, in fact, nonequivalent in topological structures. Of particular interest is the fact that the chaotic system has a chaotic attractor, one saddle and two stable node-foci. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcations, routes to chaos, periodic windows, possible chaotic and periodic-window parameter regions, and the compound structure of the system are analyzed and demonstrated with careful numerical simulations.

scholarly journals Zero–Hopf Bifurcations in Three-Dimensional Chaotic Systems with One Stable Equilibrium

2020 ◽  
Vol 30 (13) ◽  
pp. 2050189
Author(s):  
Jaume Llibre ◽  
Marcelo Messias ◽  
Alisson de Carvalho Reinol
Keyword(s):  
Equilibrium Point ◽  
Periodic Orbits ◽  
Quadratic Differential ◽  
Stable Equilibrium ◽  
Three Dimensional ◽  
Period Doubling ◽  
Stable Equilibrium Point ◽  
Differential Systems ◽  
Saddle Type ◽  
Route To Chaos

In [Molaie et al., 2013] the authors provided the expressions of 23 quadratic differential systems in [Formula: see text] with the unusual feature of having chaotic dynamics coexisting with one stable equilibrium point. In this paper, we consider 23 classes of quadratic differential systems in [Formula: see text] depending on a real parameter [Formula: see text], which, for [Formula: see text], coincide with the differential systems given by [Molaie et al., 2013]. We study the dynamics and bifurcations of these classes of differential systems by varying the parameter value [Formula: see text]. We prove that, for [Formula: see text], all the 23 considered systems have a nonisolated zero–Hopf equilibrium point located at the origin. By using the averaging theory of first order, we prove that a zero–Hopf bifurcation takes place at this point for [Formula: see text], which leads to the creation of three periodic orbits bifurcating from it for [Formula: see text] small enough: an unstable one and a pair of saddle type periodic orbits, that is, periodic orbits with a stable and an unstable manifold. Furthermore, we numerically show that the hidden chaotic attractors which exist for these systems when [Formula: see text] are obtained by period-doubling route to chaos.

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