COEXISTENCE OF POINT, PERIODIC AND STRANGE ATTRACTORS

2013 ◽  
Vol 23 (05) ◽  
pp. 1350093 ◽  
Author(s):  
JULIEN CLINTON SPROTT ◽  
XIONG WANG ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Strange Attractor ◽  
Chaotic Attractor ◽  
Stable Equilibrium ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
New Chaotic System ◽  
Point Attractor ◽  
New System

For a dynamical system described by a set of autonomous ordinary differential equations, an attractor can be a point, a periodic cycle, or even a strange attractor. Recently, a new chaotic system with only one stable equilibrium was described, which locally converges to the stable equilibrium but is globally chaotic. This paper further shows that for certain parameters, besides the point attractor and chaotic attractor, this system also has a coexisting stable limit cycle, demonstrating that this new system is truly complicated and interesting.

scholarly journals The Holling Type II Population Model Subjected to Rapid Random Attacks of Predator

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Jevgeņijs Carkovs ◽  
Jolanta Goldšteine ◽  
Kārlis Šadurskis
Keyword(s):  
Dynamical System ◽  
Population Model ◽  
Stable Limit Cycle ◽  
Time Interval ◽  
Stable Limit ◽  
Type Ii ◽  
Time Intervals ◽  
Average Motion ◽  
Trophic Function ◽  
Random Time Interval

We present the analysis of a mathematical model of the dynamics of interacting predator and prey populations with the Holling type random trophic function under the assumption of random time interval passage between predator attacks on prey. We propose a stochastic approximation algorithm for quantitative analysis of the above model based on the probabilistic limit theorem. If the predators’ gains and the time intervals between predator attacks are sufficiently small, our proposed method allows us to derive an approximative average dynamical system for mathematical expectations of population dynamics and the stochastic Ito differential equation for the random deviations from the average motion. Assuming that the averaged dynamical system is the classic Holling type II population model with asymptotically stable limit cycle, we prove that the dynamics of stochastic model may be approximated with a two-dimensional Gaussian Markov process with unboundedly increasing variances.

A New Chaotic Attractor Around a Pre-Located Ring

2017 ◽  
Vol 27 (10) ◽  
pp. 1750152 ◽  
Author(s):  
Zhen Wang ◽  
Zhe Xu ◽  
Ezzedine Mliki ◽  
Akif Akgul ◽  
Viet-Thanh Pham ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Specific Property ◽  
Unique Property ◽  
Circuit Implementation ◽  
New Chaotic System ◽  
New Chaotic Attractor ◽  
New System

Designing chaotic systems with specific features is a very interesting topic in nonlinear dynamics. However most of the efforts in this area are about features in the structure of the equations, while there is less attention to features in the topology of strange attractors. In this paper, we introduce a new chaotic system with unique property. It has been designed in such a way that a specific property has been injected to it. This new system is analyzed carefully and its real circuit implementation is presented.

Finding Alternatives and Reduced Formulations for Process-Based Models

2008 ◽  
Vol 16 (1) ◽  
pp. 63-88 ◽  
Author(s):  
Knut Bernhardt
Keyword(s):  
Differential Equations ◽  
Stable Limit Cycle ◽  
Model Systems ◽  
Model Complexity ◽  
Stable Limit ◽  
State Variables ◽  
Predator Prey ◽  
Oscillatory Dynamics ◽  
Process Based Models ◽  
Simple Models

This paper addresses the problem of model complexity commonly arising in constructing and using process-based models with intricate interactions. Apart from complex process details the dynamic behavior of such systems is often limited to a discrete number of typical states. Thus, models reproducing the system's processes in all details are often too complex and over-parameterized. In order to reduce simulation times and to get a better impression of the important mechanisms, simplified formulations are desirable. In this work a data adaptive model reduction scheme that automatically builds simple models from complex ones is proposed. The method can be applied to the transformation and reduction of systems of ordinary differential equations. It consists of a multistep approach using a low dimensional projection of the model data followed by a Genetic Programming/Genetic Algorithm hybrid to evolve new model systems. As the resulting models again consist of differential equations, their process-based interpretation in terms of new state variables becomes possible. Transformations of two simple models with oscillatory dynamics, simulating a mathematical pendulum and predator-prey interactions respectively, serve as introductory examples of the method's application. The resulting equations of force indicate the predator-prey system's equivalence to a nonlinear oscillator. In contrast to the simple pendulum it contains driving and damping forces that produce a stable limit cycle.

scholarly journals Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

2021 ◽  
Author(s):  
Vasiliy Belozyorov ◽  
Danylo Dantsev
Keyword(s):  
Time Series ◽  
Continuous Function ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Activation Functions ◽  
Kolmogorov Theorem ◽  
Stable Limit Cycles

Abstract The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. In addition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given.

scholarly journals Estimation of oscillation velocities of oscillator network

2018 ◽  
Vol 32 ◽  
pp. 182-189
Author(s):  
Volodymyr Shcherbak ◽  
Iryna Dmytryshyn
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Nonlinear Dynamical System ◽  
Stable Limit Cycle ◽  
Nonlinear Observer ◽  
Stable Limit ◽  
Nonlinear Dynamical ◽  
Biomechanical Models ◽  
Neural Ensembles

The study of the collective behavior of multiscale dynamic processes is currently one of the most urgent problems of nonlinear dynamics. Such systems arise on modelling of many cyclical biological or physical processes. It is of fundamental importance for understanding the basic laws of synchronous dynamics of distributed active subsystems with oscillations, such as neural ensembles, biomechanical models of cardiac or locomotor activity, models of turbulent media, etc. Since the nonlinear oscillations that are observed in such systems have a stable limit cycle , which does not depend on the initial conditions, then a system of interconnected nonlinear oscillators is usually used as a model of multiscale processes. The equations of Lienar type are often used as the main dynamic model of each of these oscillators. In a number of practical control problems of such interconnected oscillators it is necessary to determine the oscillation velocities by known data. This problem is considered as observation problem for nonlinear dynamical system. A new method – a synthesis of invariant relations is used to design a nonlinear observer. The method allows us to represent unknowns as a function of known quantities. The scheme of the construction of invariant relations consists in the expansion of the original dynamical system by equations of some controlled subsystem (integrator). Control in the additional system is used for the synthesis of some relations that are invariant for the extended system and have the attraction property for all of its trajectories. Such relations are considered in observation problems as additional equations for unknown state vector of initial oscillators ensemble. To design the observer, first we introduce a observer for unique oscillator of Lienar type and prove its exponential convergence. This observer is then extended on several coupled Lienar type oscillators. The performance of the proposed method is investigated by numerical simulations.

scholarly journals Design of a New Chaotic System Based on Van Der Pol Oscillator and Its Encryption Application

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 743 ◽  
Author(s):  
Jianbin He ◽  
Jianping Cai
Keyword(s):  
Control Method ◽  
Variable Parameter ◽  
Stable Limit Cycle ◽  
Simple Algorithm ◽  
Van Der Pol Oscillator ◽  
Stable Limit ◽  
Van Der Pol ◽  
Dynamical Characteristics ◽  
New Chaotic System ◽  
Exclusive Or

The Van der Pol oscillator is investigated by the parameter control method. This method only needs to control one parameter of the Van der Pol oscillator by a simple periodic function; then, the Van der Pol oscillator can behave chaotically from the stable limit cycle. Based on the new Van der Pol oscillator with variable parameter (VdPVP), some dynamical characteristics are discussed by numerical simulations, such as the Lyapunov exponents and bifurcation diagrams. The numerical results show that there exists a positive Lyapunov exponent in the VdPVP. Therefore, an encryption algorithm is designed by the pseudo-random sequences generated from the VdPVP. This simple algorithm consists of chaos scrambling and chaos XOR (exclusive-or) operation, and the statistical analyses show that it has good security and encryption effectiveness. Finally, the feasibility and validity are verified by simulation experiments of image encryption.

scholarly journals A New Four-Scroll Chaotic Attractor Consisted of Two-Scroll Transient Chaotic and Two-Scroll Ultimate Chaotic

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Yuhua Xu ◽  
Bing Li ◽  
Yuling Wang ◽  
Wuneng Zhou ◽  
Jian-an Fang
Keyword(s):  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
The Other ◽  
Phase Portraits ◽  
Fractional Dimension ◽  
The Novel ◽  
New Chaotic System ◽  
New System

A new four-scroll chaotic attractor is found by feedback controlling method in this paper. The novel chaotic system can generate four scrolls two of which are transient chaotic and the other two of which are ultimate chaotic. Of particular interest is that this novel chaotic system can generate one-scroll, two 2-scroll and four-scroll chaotic attractor with variation of a single parameter. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map, respectively. The analysis results show clearly that this is a new chaotic system which deserves further detailed investigation.

scholarly journals Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Marluci Cristina Galindo ◽  
Cristiane Nespoli ◽  
Marcelo Messias
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Chaotic Attractor ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Tumour Cells ◽  
Dimensional System ◽  
Stable Limit ◽  
Cancer Model

We study a cancer model given by a three-dimensional system of ordinary differential equations, depending on eight parameters, which describe the interaction among healthy cells, tumour cells, and effector cells of immune system. The model was previously studied in the literature and was shown to have a chaotic attractor. In this paper we study how such a chaotic attractor is formed. More precisely, by varying one of the parameters, we prove that a supercritical Hopf bifurcation occurs, leading to the creation of a stable limit cycle. Then studying the continuation of this limit cycle we numerically found a cascade of period-doubling bifurcations which leads to the formation of the mentioned chaotic attractor. Moreover, analyzing the model dynamics from a biological point of view, we notice the possibility of both the tumour cells and the immune system cells to vanish and only the healthy cells survive, suggesting the possibility of cure, since the interactions with the immune system can eliminate tumour cells.

COHERENT SYNCHRONIZATION IN LINEARLY COUPLED NONLINEAR SYSTEMS

2006 ◽  
Vol 16 (05) ◽  
pp. 1375-1387 ◽  
Author(s):  
TIANSHOU ZHOU ◽  
GUANRONG CHEN
Keyword(s):  
Limit Cycle ◽  
Coupling Strength ◽  
Chaotic Attractor ◽  
Chaotic Behavior ◽  
Quantitative Description ◽  
Stable Limit Cycle ◽  
Coupled System ◽  
Coupled Systems ◽  
Stable Limit ◽  
Single System

This paper presents a novel result on the effect of coupling through both analytical and numerical investigations on linearly coupled systems including chaotic and nonchaotic systems. It is found that when a single system has potential of oscillation but is currently in a "marginal" state to produce a limit cycle via Hopf bifurcation due to the change of a parameter, an appropriate coupling strength can excite the potential limit cycle such that the coupled system oscillates synchronously. Similarly, when a stable limit cycle is at the "margin" of a chaotic attractor in a single system, a certain coupling strength can induce the potential chaotic attractor such that the coupled system has a synchronous chaotic behavior. This excitation mechanism is different from the traditional function of coupling in that the latter mainly drives the coupled system to synchronize with the ongoing dynamics of a single system but does not recover its disappearing dynamics. This newly observed synchronization is called coherent synchronization to distinguish it from various common types of synchronization. Several numerical examples are presented for quantitative description of this interesting phenomenon.

Sign in / Sign up

Export Citation Format

Share Document