MAKING A DISCRETE DYNAMICAL SYSTEM CHAOTIC: THEORETICAL RESULTS AND NUMERICAL SIMULATIONS

2003 ◽  
Vol 13 (11) ◽  
pp. 3437-3442 ◽  
Author(s):  
DEJIAN LAI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Computer Simulations ◽  
State Feedback ◽  
Controller Design ◽  
Discrete Dynamical System ◽  
Finite Samples ◽  
Dimensional Dynamical System ◽  
Theoretical Results

In this paper, we study state-feedback controller design for controlling the Lyapunov exponents of an n-dimensional dynamical system. We examine some theoretical results and perform numerical simulations for systems with and without noise influence. The controlled Lyapunov exponents are asymptotically normally distributed if the system has noisy inputs. Computer simulations on finite samples are all consistent with the theoretical results.

Approximating Lyapunov Exponents of Discrete Dynamical Systems

2019 ◽  
Vol 11 (11) ◽  
pp. 1612-1615
Author(s):  
Wadia Faid Hassan Al-Shameri
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Science And Technology ◽  
Discrete Dynamical System ◽  
Discrete Dynamical Systems ◽  
Significant Part ◽  
Matlab Code

Lyapunov exponents play a significant part in revealing and quantifying chaos, which occurs in many areas of science and technology. The purpose of this study was to approximate the Lyapunov exponents for discrete dynamical systems and to present it as a quantifier for inferring and detecting the existence of chaos in those discrete dynamical systems. Finally, the approximation of the Lyapunov exponents for the discrete dynamical system was implemented using the Matlab code listed in the Appendix.

scholarly journals Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zongcheng Li
Keyword(s):  
Dynamical System ◽  
Computer Simulations ◽  
Difference Equations ◽  
Discrete Dynamical System ◽  
Control Technique ◽  
High Dimensional ◽  
Controlled System ◽  
Heteroclinic Cycles ◽  
Delay Difference Equations ◽  
Delay Difference

This paper is concerned with anticontrol of chaos for a class of delay difference equations via the feedback control technique. The controlled system is first reformulated into a high-dimensional discrete dynamical system. Then, a chaotification theorem based on the heteroclinic cycles connecting repellers for maps is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations.

scholarly journals Controlling hopf bifurcations: Discrete-time systems

2000 ◽  
Vol 5 (1) ◽  
pp. 29-33 ◽  
Author(s):  
Guanrong Chen ◽  
Jin-quing Fang ◽  
Yiguang Hong ◽  
Huashu Qin
Keyword(s):  
Hopf Bifurcation ◽  
Computer Simulations ◽  
Discrete Time ◽  
State Feedback ◽  
Bifurcation Control ◽  
Hopf Bifurcations ◽  
Control Task ◽  
Discrete Time Systems ◽  
Time Systems ◽  
Theoretical Results

Bifurcation control has attracted increasing attention in recent years. A simple and unified state-feedback methodology is developed in this paper for Hopf bifurcation control for discrete-time systems. The control task can be either shifting an existing Hopf bifurcation or creating a new Hopf bifurcation. Some computer simulations are included to illustrate the methodology and to verify the theoretical results.

QUANTIZATION EFFECT ON A SECOND-ORDER DYNAMICAL SYSTEM UNDER SLIDING-MODE CONTROL

2013 ◽  
Vol 23 (08) ◽  
pp. 1350131 ◽  
Author(s):  
YAN YAN ◽  
XINGHUO YU ◽  
SHUANGHE YU
Keyword(s):  
Dynamical System ◽  
Control System ◽  
Sliding Mode Control ◽  
State Feedback ◽  
Sliding Mode ◽  
The State ◽  
Quantization Effect ◽  
Mode Control ◽  
Dynamical Properties ◽  
Theoretical Results

The quantization behaviors of a sliding-mode control system with state feedback when the measurement of the state is quantized are studied. Some inherent dynamical properties for the quantized SMC systems are analyzed. The definition and the occurrence condition of "quantized sliding mode" are proposed. Theoretical results are illustrated with simulation examples.

scholarly journals A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative

2021 ◽  
Vol 5 (3) ◽  
pp. 87
Author(s):  
Youcef Belgaid ◽  
Mohamed Helal ◽  
Abdelkader Lakmeche ◽  
Ezio Venturino
Keyword(s):  
Dynamical System ◽  
Fractional Derivative ◽  
Numerical Simulations ◽  
Minimal Model ◽  
Fractional Operators ◽  
Mathematical Study ◽  
Stability Behavior ◽  
Relevant Role ◽  
Caputo Fractional Order Derivative ◽  
Theoretical Results

In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate that their stability behavior is the same as for the corresponding system formulated via standard derivatives. This suggests that, at least in this case for the model presented here, the memory effects contained in the fractional operators apparently do not seem to play a relevant role. The numerical simulations instead reveal that the order of the fractional derivative has a definite influence on both the equilibrium population levels and the speed at which they are attained.

scholarly journals Dynamics of a System Associated with a Piecewise Quadratic Family

2016 ◽  
Vol 7 (2) ◽  
pp. 125
Author(s):  
KAREN LOPEZ BURITICA ◽  
Simeón Casanova Trujillo ◽  
Carlos Daniel Acosta Medina
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponents ◽  
Discrete Dynamical System ◽  
Period Doubling ◽  
Quadratic Family

AbstractThis paper presents a study, both in analytical and numerical form, of a discrete dynamical system associated with a piecewise quadratic family. The orbits of periods one and two are characterized, and their stability is established. The nonsmooth phenomenon known as border collision is present when there is a period doubling. Lyapunov exponents are calculated numerically to determine the presence of chaos in the system.

HYBRID CONTROL OF BIFURCATION IN CONTINUOUS NONLINEAR DYNAMICAL SYSTEMS

2005 ◽  
Vol 15 (12) ◽  
pp. 3895-3903 ◽  
Author(s):  
ZENGRONG LIU ◽  
K. W. CHUNG
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Control Strategy ◽  
State Feedback ◽  
Hybrid Control ◽  
Nonlinear Dynamical Systems ◽  
Parameter Perturbation ◽  
Nonlinear Dynamical ◽  
Dimensional Dynamical System

In this paper, a new hybrid control strategy is proposed, in which state feedback and parameter perturbation are used to control the bifurcations of continuous dynamical systems. The hybrid control can be applied to any component of a several dimensional dynamical system and is still effective even when the system becomes chaotic. Our results show that various bifurcations, such as Hopf bifurcation and Poincaré bifurcation, can be controlled by means of this method.

scholarly journals l1-Induced State-Feedback Controller Design for Positive Fuzzy Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Xiaoming Chen ◽  
Mou Chen ◽  
Jun Shen
Keyword(s):  
Fuzzy Systems ◽  
State Feedback ◽  
Closed Loop ◽  
Controller Design ◽  
Sufficient Conditions ◽  
Feedback Controller ◽  
Closed Loop System ◽  
State Feedback Controller ◽  
Takagi Sugeno ◽  
Theoretical Results

The problem ofl1-induced state-feedback controller design is investigated for positive Takagi-Sugeno (T-S) fuzzy systems with the use of linear Lyapunov function. First, a novel performance characterization is established to guarantee the asymptotic stability of the closed-loop system withl1-induced performance. Then, the sufficient conditions are presented to design the required fuzzy controllers and iterative convex optimization approaches are developed to solve the conditions. Finally, one example is presented to show the effectiveness of the derived theoretical results.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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