scholarly journals Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Aiguo Wu ◽  
Shijian Cang ◽  
Ruiye Zhang ◽  
Zenghui Wang ◽  
Zengqiang Chen
Keyword(s):  
Fixed Points ◽  
Chaotic Dynamics ◽  
Conservative System ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Numerical Techniques ◽  
Natural Systems ◽  
Weather And Climate ◽  
Conservative Systems ◽  
Perpetual Points

Chaotic dynamics exists in many natural systems, such as weather and climate, and there are many applications in different disciplines. However, there are few research results about chaotic conservative systems especially the smooth hyperchaotic conservative system in both theory and application. This paper proposes a five-dimensional (5D) smooth autonomous hyperchaotic system with nonhyperbolic fixed points. Although the proposed system includes four linear terms and four quadratic terms, the new system shows complicated dynamics which has been proven by the theoretical analysis. Several notable properties related to conservative systems and the existence of perpetual points are investigated for the proposed system. Moreover, its conservative hyperchaotic behavior is illustrated by numerical techniques including phase portraits and Lyapunov exponents.

Stability of equilibria and fixed points of conservative systems

Nonlinearity ◽  
1999 ◽  
Vol 12 (5) ◽  
pp. 1351-1362 ◽  
Author(s):  
Hildeberto E Cabral ◽  
Kenneth R Meyer
Keyword(s):  
Fixed Points ◽  
Stability Of Equilibria ◽  
Conservative Systems

scholarly journals Chaos-Based Application of a Novel Multistable 5D Memristive Hyperchaotic System with Coexisting Multiple Attractors

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19 ◽  
Author(s):  
Fei Yu ◽  
Li Liu ◽  
Shuai Qian ◽  
Lixiang Li ◽  
Yuanyuan Huang ◽  
...  
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Random Number Generator ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
The Novel ◽  
Multiple Attractors ◽  
Period 2 ◽  
Different Types ◽  
System Designs

Novel memristive hyperchaotic system designs and their engineering applications have received considerable critical attention. In this paper, a novel multistable 5D memristive hyperchaotic system and its application are introduced. The interesting aspect of this chaotic system is that it has different types of coexisting attractors, chaos, hyperchaos, periods, and limit cycles. First, a novel 5D memristive hyperchaotic system is proposed by introducing a flux-controlled memristor with quadratic nonlinearity into an existing 4D four-wing chaotic system as a feedback term. Then, the phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy are used to analyze the basic dynamics of the 5D memristive hyperchaotic system. For a specific set of parameters, we find an unusual metastability, which shows the transition from chaotic to periodic (period-2 and period-3) dynamics. Moreover, its circuit implementation is also proposed. By using the chaoticity of the novel hyperchaotic system, we have developed a random number generator (RNG) for practical image encryption applications. Furthermore, security analyses are carried out with the RNG and image encryption designs.

Dynamical Analysis and Circuit Simulation of a New Fractional-Order Hyperchaotic System and Its Discretization

2016 ◽  
Vol 26 (13) ◽  
pp. 1650222 ◽  
Author(s):  
A. M. A. El-Sayed ◽  
A. Elsonbaty ◽  
A. A. Elsadany ◽  
A. E. Matouk
Keyword(s):  
Fractional Order ◽  
Initial Conditions ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Control Technique ◽  
Order System ◽  
Qualitative Behavior ◽  
Hyperchaotic System

This paper presents an analytical framework to investigate the dynamical behavior of a new fractional-order hyperchaotic circuit system. A sufficient condition for existence, uniqueness and continuous dependence on initial conditions of the solution of the proposed system is derived. The local stability of all the system’s equilibrium points are discussed using fractional Routh–Hurwitz test. Then the analytical conditions for the existence of a pitchfork bifurcation in this system with fractional-order parameter less than 1/3 are provided. Conditions for the existence of Hopf bifurcation in this system are also investigated. The dynamics of discretized form of our fractional-order hyperchaotic system are explored. Chaos control is also achieved in discretized system using delay feedback control technique. The numerical simulation are presented to confirm our theoretical analysis via phase portraits, bifurcation diagrams and Lyapunov exponents. A text encryption algorithm is presented based on the proposed fractional-order system. The results show that the new system exhibits a rich variety of dynamical behaviors such as limit cycles, chaos and transient phenomena where fractional-order derivative represents a key parameter in determining system qualitative behavior.

Analysis of the Chaotic Dynamics of MEMS/NEMS Doubly Clamped Beam Resonators with Two-Sided Electrodes

2018 ◽  
Vol 28 (10) ◽  
pp. 1850122 ◽  
Author(s):  
W. G. Dantas ◽  
A. Gusso
Keyword(s):  
Differential Equation ◽  
Chaotic Dynamics ◽  
Electrostatic Force ◽  
Beam Theory ◽  
Nonlinear Ordinary Differential Equation ◽  
Phase Portraits ◽  
Physical Parameters ◽  
Relevant Parameter ◽  
Practical Applications ◽  
Beam Curvature

We investigate the chaotic dynamics of micro- and nanoelectromechanical (MEMS/NEMS) beam resonators actuated electrostatically by two-sided electrodes, considering devices with realistic physical parameters. We model the resonators using the Euler–Bernoulli beam theory with the addition of viscous damping, midplane stretching and the electrostatic force. For the purpose of numerical simulations, the partial differential equation describing the system is reduced to a one degree of freedom model using the Galerkin method. The resulting nonlinear ordinary differential equation incorporates the main effects of the beam curvature. A comparison with the widely used parallel plate approximation (PPA) evidences the significant effects of the beam curvature. It is also concluded that in the case of resonators with two-sided electrodes special care must be taken when using the PPA. A detailed numerical analysis reveals the region in the relevant parameter space where chaos can be found. Phase portraits, Poincaré sections and bifurcation diagrams are used to characterize the chaotic attractors. The effects of gap asymmetry and damping are also investigated, showing that a stronger chaotic dynamics is favored by small asymmetries and smaller damping. In general, a more complex chaotic dynamics was found, compared to what was initially expected. The results are relevant in view of the potential practical applications in the generation of pseudo-random numbers and chaotic signals for secure communications. The proposed improved model can be easily implemented numerically, helping in the design and simulation of resonators, and the comparison between theoretical and experimental results.

scholarly journals Why many theories of shock waves are necessary: kinetic relations for non-conservative systems

2012 ◽  
Vol 142 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Christophe Berthon ◽  
Frédéric Coquel ◽  
Philippe G. LeFloch
Keyword(s):  
Shock Waves ◽  
Hyperbolic Systems ◽  
Travelling Waves ◽  
Conservative System ◽  
Travelling Wave ◽  
Phase Plane Analysis ◽  
Kinetic Relation ◽  
Lipschitz Continuous ◽  
Entropy Dissipation ◽  
Conservative Systems

For a class of non-conservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial-value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to non-conservative systems of a similar concept introduced by Abeyaratne, Knowles and Truskinovsky for subsonic phase transitions and by LeFloch for non-classical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for non-conservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase-plane analysis of travelling-wave solutions associated with an augmented version of the non-conservative system. We illustrate with several examples that non-conservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of travelling waves which yields the corresponding kinetic function.

THE GENERATION AND ANALYSIS OF A NEW FOUR-DIMENSIONAL HYPERCHAOTIC SYSTEM

2007 ◽  
Vol 18 (06) ◽  
pp. 1013-1024 ◽  
Author(s):  
JIEZHI WANG ◽  
ZENGQIANG CHEN ◽  
ZHUZHI YUAN
Keyword(s):  
Lyapunov Exponent ◽  
Cross Product ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Bifurcation Diagrams ◽  
Dynamic Behaviors ◽  
System Parameters ◽  
Product Term ◽  
Two Parameters ◽  
Cross Product Term

A new four-dimensional continuous autonomous hyperchaotic system is considered. It possesses two parameters, and each equation of it has one quadratic cross product term. Some basic properties of it are studied. The dynamic behaviors of it are analyzed by the Lyapunov exponent (LE) spectrum, bifurcation diagrams, phase portraits, and Poincaré sections. The system has larger hyperchaotic region. When it is hyperchaotic, the two positive LE are both large and they are both larger than 1 if the system parameters are taken appropriately.

scholarly journals Theoretical Analysis and Circuit Verification for Fractional-Order Chaotic Behavior in a New Hyperchaotic System

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ling Liu ◽  
Chongxin Liu
Keyword(s):  
Fractional Order ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Circuit Diagram ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
Dynamical Behaviors ◽  
Mapping Parameter ◽  
New Hyperchaotic System ◽  
Observation Results

A novel nonlinear four-dimensional hyperchaotic system and its fractional-order form are presented. Some dynamical behaviors of this system are further investigated, including Poincaré mapping, parameter phase portraits, equilibrium points, bifurcations, and calculated Lyapunov exponents. A simple fourth-channel block circuit diagram is designed for generating strange attractors of this dynamical system. Specifically, a novel network module fractance is introduced to achieve fractional-order circuit diagram for hardware implementation of the fractional attractors of this nonlinear hyperchaotic system with order as low as 0.9. Observation results have been observed by using oscilloscope which demonstrate that the fractional-order nonlinear hyperchaotic attractors exist indeed in this new system.

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.

Sign in / Sign up

Export Citation Format

Share Document