Symbolic Dynamics in Multidimensional Annulus and Chimera States

2014 ◽  
Vol 24 (08) ◽  
pp. 1440002
Author(s):  
V. S. Afraimovich ◽  
L. P. Shilnikov
Keyword(s):  
Symbolic Dynamics ◽  
Chaotic Behavior ◽  
Sufficient Conditions ◽  
Invariant Set ◽  
Chimera States ◽  
Chimera State ◽  
State Type ◽  
Symbolic Sequences ◽  
Angular Coordinates ◽  
Low Dimensional

We derive sufficient conditions for the existence of an invariant set in an absorbing region homeomorphic to the product of a multidimensional torus and a ball. This set consists of low dimensional tori labeled by symbolic sequences. It may appear as a result of the breakdown of an attracting multidimensional torus. Trajectories on the set manifest chaotic behavior for some angular coordinates and may behave regularly for others, i.e. the dynamics on the set is of the chimera state type.

Impulsive Control and Synchronization of Memristor-Based Chaotic Circuits

2014 ◽  
Vol 24 (12) ◽  
pp. 1450162 ◽  
Author(s):  
Shiju Yang ◽  
Chuandong Li ◽  
Tingwen Huang
Keyword(s):  
Chaotic System ◽  
Impulsive Control ◽  
Chaotic Behavior ◽  
Memory Function ◽  
Sufficient Conditions ◽  
Impulsive System ◽  
Circuit Element ◽  
Chaotic Circuits ◽  
Control And Synchronization ◽  
Passive Circuit

The memristor is a novel nonlinear passive circuit element which has the memory function, and the circuits based on the memristors might exhibit chaotic behavior. In this paper, we revisit a memristor-based chaotic circuit, and then investigate its stabilization and synchronization via impulsive control. By impulsive system theory, some sufficient conditions for the stabilization and synchronization of the memristor-based chaotic system are established. Moreover, an estimation of the upper bound of the impulse interval is proposed under the condition that the parameters of the chaotic system and the impulsive control law are well defined. To show the effectiveness of the theoretical results, numerical simulations are also presented.

scholarly journals Chaotic Behavior of One-Dimensional Cellular Automata Rule 24

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zujie Bie ◽  
Qi Han ◽  
Chao Liu ◽  
Junjian Huang ◽  
Lepeng Song ◽  
...  
Keyword(s):  
Cellular Automata ◽  
Characteristic Function ◽  
Symbolic Dynamics ◽  
Bernoulli Shift ◽  
Chaotic Behavior ◽  
Class Ii ◽  
Chaotic Attractors ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
Shift Map

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 24, which is Bernoulliστ-shift rule and is member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of four rules, whether they possess chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 24 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 24 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Furthermore, we prove that four rules of global equivalenceε52of cellular automata are topologically conjugate. Then, we use diagrams to explain the attractor of rule 24, where characteristic function is used to describe the fact that all points fall into Bernoulli-shift map after two iterations under rule 24.

Stability of a Mathematical Model with Piecewise Constant Arguments for Tumor-Immune Interaction Under Drug Therapy

2019 ◽  
Vol 29 (01) ◽  
pp. 1950009 ◽  
Author(s):  
Zonghong Feng ◽  
Xinxing Wu ◽  
Luo Yang
Keyword(s):  
Mathematical Model ◽  
Growth Rate ◽  
Drug Therapy ◽  
Tumor Growth ◽  
Tumor Cells ◽  
Chaotic Behavior ◽  
Sufficient Conditions ◽  
Tumor Growth Rate ◽  
Chaotic Region ◽  
Treatment Parameter

This paper studies a mathematical model for the interaction between tumor cells and Cytotoxic T lymphocytes (CTLs) under drug therapy. We obtain some sufficient conditions for the local and global asymptotical stabilities of the system by using Schur–Cohn criterion and the theory of Lyapunov function. In addition, it is known that the system without any treatment may undergo Neimark–Sacker bifurcation, and there may exist a chaotic region of values of tumor growth rate where the system exhibits chaotic behavior. So it is important to narrow the chaotic region. This may be done by increasing the intensity of the treatment to some extent. Moreover, for a fixed value of tumor growth rate in the chaotic region, a threshold value [Formula: see text] is predicted of the treatment parameter [Formula: see text]. We can see Neimark–Sacker bifurcation of the system when [Formula: see text], and the chaotic behavior for tumor cells ends and the system becomes locally asymptotically stable when [Formula: see text].

Periodic orbits analysis for the Zhou system via variational approach

2019 ◽  
Vol 33 (19) ◽  
pp. 1950212 ◽  
Author(s):  
Chengwei Dong ◽  
Lian Jia
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Variational Approach ◽  
Topological Classification ◽  
Dissipative Systems ◽  
Systematic Calculation ◽  
Low Dimensional ◽  
General Method

We proposed a general method for the systematic calculation of unstable cycles in the Zhou system. The variational approach is employed for the cycle search, and we establish interesting symbolic dynamics successfully based on the orbits circuiting property with respect to different fixed points. Upon the defined symbolic rule, cycles with topological length up to five are sought and ordered. Further, upon parameter changes, the homotopy evolution of certain selected cycles are investigated. The topological classification methodology could be widely utilized in other low-dimensional dissipative systems.

Lyapunov-Based Sufficient Conditions for Nonlinear Impulsive Systems

2011 ◽  
Vol 219-220 ◽  
pp. 508-512
Author(s):  
Yong Liang Gao ◽  
Xiao Wu Mu
Keyword(s):  
Differential Equation ◽  
Stability Analysis ◽  
Lyapunov Function ◽  
Sufficient Conditions ◽  
Invariant Set ◽  
Positive Definite ◽  
Impulsive Systems ◽  
Stability Theorems ◽  
Sufficient Conditions For Convergence ◽  
The Stability

This paper focuses on the stability analysis and invariant set stability theorems for nonlinear impulsive systems. A set of Lyapunov-based sufficient conditions are established for these convergent properties. These results do not require the Lyapunov function to be positive definite. Inequalities relating the righthandside of the differential equation and the Lyapunov function derivative are involved for these results. These inequalities make it possible to deduce properties of the functions and thus leads to sufficient conditions for convergence and stability.

scholarly journals Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four-Dimensional Hyperchaotic Financial System

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. C. Wei ◽  
J. F. Wang ◽  
A. Akgul
Keyword(s):  
Hopf Bifurcation ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Financial System ◽  
Invariant Set ◽  
Phase Portraits ◽  
Lyapunov Coefficient ◽  
Stable Periodic ◽  
Analytical Proof ◽  
Positively Invariant Set

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

GLOBALLY EXPONENTIALLY ATTRACTIVE SETS AND POSITIVE INVARIANT SETS OF THREE-DIMENSIONAL AUTONOMOUS SYSTEMS WITH ONLY CROSS-PRODUCT NONLINEARITIES

2013 ◽  
Vol 23 (01) ◽  
pp. 1350007 ◽  
Author(s):  
XINQUAN ZHAO ◽  
FENG JIANG ◽  
JUNHAO HU
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Cross Product ◽  
Invariant Set ◽  
Invariant Sets ◽  
Special Cases ◽  
Attractive Sets ◽  
Positive Invariant Set

In this paper, the existence of globally exponentially attractive sets and positive invariant sets of three-dimensional autonomous systems with only cross-product nonlinearities are considered. Sufficient conditions, which guarantee the existence of globally exponentially attractive set and positive invariant set of the system, are obtained. The results of this paper comprise some existing relative results as in special cases. The approach presented in this paper can be applied to study other chaotic systems.

scholarly journals Cascades of Multiheaded Chimera States for Coupled Phase Oscillators

2014 ◽  
Vol 24 (08) ◽  
pp. 1440014 ◽  
Author(s):  
Yuri L. Maistrenko ◽  
Anna Vasylenko ◽  
Oleksandr Sudakov ◽  
Roman Levchenko ◽  
Volodymyr L. Maistrenko
Keyword(s):  
Periodic Orbit ◽  
Coupled Oscillators ◽  
Phase Oscillators ◽  
Invariant Circle ◽  
Saddle Node Bifurcation ◽  
Chimera States ◽  
Chimera State ◽  
Self Organized ◽  
Saddle Node ◽  
Different Types

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

scholarly journals Dynamic Analysis of an SEIR Model with Distinct Incidence for Exposed and Infectives

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Junhong Li ◽  
Ning Cui
Keyword(s):  
Matrix Theory ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Vaccination Strategy ◽  
Invariant Set ◽  
Incidence Rates ◽  
Seir Model ◽  
Asymptotical Stable ◽  
Compound Matrix ◽  
Disease Free

An SEIR model with vaccination strategy that incorporates distinct incidence rates for the exposed and the infected populations is studied. By means of Lyapunov function and LaSalle’s invariant set theorem, we proved the global asymptotical stable results of the disease-free equilibrium. The sufficient conditions for the global stability of the endemic equilibrium are obtained using the compound matrix theory. Furthermore, the method of direct numerical simulation of the system shows that there is a periodic solution, when the system has three equilibrium points.

scholarly journals Braided Endomorphisms of Cuntz Algebras

2000 ◽  
Vol 87 (1) ◽  
pp. 93 ◽  
Author(s):  
Roberto Conti ◽  
Francesco Fidaleo
Keyword(s):  
Sufficient Conditions ◽  
Matrix Group ◽  
Quantum Field ◽  
Fusion Rules ◽  
Cuntz Algebras ◽  
Root Of Unity ◽  
Fourth Root ◽  
Dimension 2 ◽  
Index 2 ◽  
Low Dimensional

We discuss sufficient conditions ensuring that certain endomorphisms of infinite factors arising from Cuntz algebras are braided. We analyse some explicit non-trivial examples associated to unitary solutions of quantum Yang-Baxter equations on a Hilbert space of dimension 2. In particular we show the existence of endomorphisms of index 2 associated to representations of Hecke algebras at a primitive fourth root of unity. In this case we compute the associated fusion rules. These fusion rules define a finitely generated *-semiring which is not finite. Such a picture seems to be closely related to the description of (the dual of) a deformation, at a fourth root of unity, of some compact matrix group. This could be of some interest for the investigation of quantum summetries naturally appearing in low-dimensional Quantum Field Theory.

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