scholarly journals A computational framework for finding parameter sets associated with chaotic dynamics

2021 ◽  
pp. 1-11
Author(s):  
S. Koshy-Chenthittayil ◽  
E. Dimitrova ◽  
E.W. Jenkins ◽  
B.C. Dean
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Computational Framework ◽  
Independent Variables ◽  
Systems Model ◽  
Small Support

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software’s optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

scholarly journals Dynamics of the Modified n-Degree Lorenz System

2019 ◽  
Vol 4 (2) ◽  
pp. 315-330 ◽  
Author(s):  
Sk. Sarif Hassan ◽  
Moole Parameswar Reddy ◽  
Ranjeet Kumar Rout
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Lyapunov Exponents ◽  
Limit Cycle ◽  
Chaotic Dynamics ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Lorenz Model ◽  
Periodic Behavior ◽  
Chaotic Model

AbstractThe Lorenz model is one of the most studied dynamical systems. Chaotic dynamics of several modified models of the classical Lorenz system are studied. In this article, a new chaotic model is introduced and studied computationally. By finding the fixed points, the eigenvalues of the Jacobian, and the Lyapunov exponents. Transition from convergence behavior to the periodic behavior (limit cycle) are observed by varying the degree of the system. Also transiting from periodic behavior to the chaotic behavior are seen by changing the degree of the system.

CHAOS AND COMPLEXITY

2001 ◽  
Vol 11 (01) ◽  
pp. 19-26 ◽  
Author(s):  
RAY BROWN ◽  
ROBERT BEREZDIVIN ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
High Dimensional ◽  
Topological Conjugacy ◽  
Integer Lattices ◽  
Homoclinic Tangles ◽  
Dimensional Complexity ◽  
Finite Integer ◽  
The Relationship

In this paper we show how to relate a form of high-dimensional complexity to chaotic and other types of dynamical systems. The derivation shows how "near-chaotic" complexity can arise without the presence of homoclinic tangles or positive Lyapunov exponents. The relationship we derive follows from the observation that the elements of invariant finite integer lattices of high-dimensional dynamical systems can, themselves, be viewed as single integers rather than coordinates of a point in n-space. From this observation it is possible to construct high-dimensional dynamical systems which have properties of shifts but for which there is no conventional topological conjugacy to a shift. The particular manner in which the shift appears in high-dimensional dynamical systems suggests that some forms of complexity arise from the presence of chaotic dynamics which are obscured by the large dimensionality of the system domain.

scholarly journals Standing Swells Surveyed Showing Surprisingly Stable Solutions for the Lorenz '96 Model

2014 ◽  
Vol 24 (10) ◽  
pp. 1430027 ◽  
Author(s):  
Morgan R. Frank ◽  
Lewis Mitchell ◽  
Peter Sheridan Dodds ◽  
Christopher M. Danforth
Keyword(s):  
Parameter Space ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Standing Waves ◽  
Nonlinear Dynamical ◽  
Hidden Order ◽  
Stable Solutions ◽  
Lorenz 96

The Lorenz '96 model is an adjustable dimension system of ODEs exhibiting chaotic behavior representative of the dynamics observed in the Earth's atmosphere. In the present study, we characterize statistical properties of the chaotic dynamics while varying the degrees of freedom and the forcing. Tuning the dimensionality of the system, we find regions of parameter space with surprising stability in the form of standing waves traveling amongst the slow oscillators. The boundaries of these stable regions fluctuate regularly with the number of slow oscillators. These results demonstrate hidden order in the Lorenz '96 system, strengthening the evidence for its role as a hallmark representative of nonlinear dynamical behavior.

Chaos in the Quasiperiodically Forced Helmholtz-Duffing Oscillator With Two-Well Potential

2005 ◽  
Author(s):  
Jing-Jun Lou ◽  
Shi-Jian Zhu ◽  
Qi-Wei He
Keyword(s):  
Hamiltonian System ◽  
Parameter Space ◽  
Homoclinic Orbit ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Restoring Force ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
The Asymmetry

The chaotic dynamics of the quasiperiodically excited Helmholtz-Duffing oscillator with two-well potential was investigated. The condition of the existence of homoclinic orbit in the corresponding Hamiltonian system was presented which is asymmetrical resulting from the asymmetry restoring force. It was found that the mechanism for chaos is transverse homoclinic tori and it is illustrated how transverse homoclinic tori give rise to chaos for the Helmholtz-Duffing oscillator with multi-frequency periodic forces. The criterion for the existence of chaos was given utilizing a generalization of the Melnikov’s method. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies increases the area in parameter space where chaotic behavior can occur.

CHAOS IN A NEAR-INTEGRABLE HAMILTONIAN LATTICE

2002 ◽  
Vol 12 (08) ◽  
pp. 1743-1754 ◽  
Author(s):  
VASSILIOS M. ROTHOS ◽  
CHRIS ANTONOPOULOS ◽  
LAMBROS DROSSOS
Keyword(s):  
Lyapunov Exponents ◽  
Numerical Experiments ◽  
Chaotic Dynamics ◽  
Large Scale ◽  
Hamiltonian Structure ◽  
Chaotic Behavior ◽  
Original System ◽  
Homoclinic Orbits ◽  
Hamiltonian Lattice ◽  
Whiskered Tori

We study the chaotic dynamics of a near-integrable Hamiltonian Ablowitz–Ladik lattice, which is N + 2-dimensional if N is even (N + 1, if N is odd) and possesses, for all N, a circle of unstable equilibria at ε = 0, whose homoclinic orbits are shown to persist for ε ≠ 0 on whiskered tori. The persistence of homoclinic orbits is established through Mel'nikov conditions, directly from the Hamiltonian structure of the equations. Numerical experiments which combine space portraits and Lyapunov exponents are performed for the perturbed Ablowitz–Ladik lattice and large scale chaotic behavior is observed in the vicinity of the circle of unstable equilibria in the ε = 0 case. We conjecture that this large scale chaos is due to the occurrence of saddle-center type fixed points in a perturbed 1 d.o.f Hamiltonian to which the original system can be reduced for all N. As ε > 0 increases, the transient character of this chaotic behavior becomes apparent as the positive Lyapunov exponents steadily increase and the orbits escape to infinity.

BISTABILITY AND DYNAMICAL TRANSITIONS IN A PHASE-MODULATED DELAYED SYSTEM

2009 ◽  
Vol 23 (23) ◽  
pp. 2733-2743 ◽  
Author(s):  
YONGXIANG ZHANG ◽  
GUIQIN KONG ◽  
JIANNING YU
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Basins Of Attraction ◽  
Parameter Study ◽  
Global Bifurcations ◽  
Coexisting Attractors ◽  
System Parameters ◽  
Different Types ◽  
Delayed System

We study a delayed system with feedback modulation of the nonlinear parameter. Study of the system as a function of nonlinearity and modulation parameters reveals complex dynamical phenomena: different types of coexisting attractors, local or global bifurcations and transitions. Bistability and dynamical attractors can be distinguished in some parameter-space regions, which may be useful to drive chaotic dynamics, unstable attractors or bistability towards regular dynamics. At the bifurcation to bistability, two striking features are that they lead to fundamentally unpredictable behavior of orbits and crisis of attractors as system parameters are varied slowly through the critical curve. Unstable attractors are also investigated in bistable regions, which are easily mistaken for true multi-periodic orbits judging merely from zero measure local basins. Lyapunov exponents and basins of attraction are also used to characterize the phenomenon observed.

CONSTANT-PHASE-INDUCED CONTROL OF CHAOTIC CHARGED PARTICLES IN THE FIELD OF A WAVE PACKET

2013 ◽  
Vol 23 (06) ◽  
pp. 1330022
Author(s):  
RICARDO CHACÓN
Keyword(s):  
Charged Particle ◽  
Wave Packet ◽  
Finite Number ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Charged Particles ◽  
Melnikov Method ◽  
Wide Range ◽  
Range Of Values

It is shown that the dissipative chaotic dynamics of a charged particle in the field of a wave packet with an arbitrary but finite number of harmonics can be reliably suppressed by judiciously varying the constant phase of the main harmonic, ϕ0, while keeping null the corresponding constant phases of the remaining harmonics. The dependence of the chaotic threshold on the wave packet parameters is predicted theoretically (Melnikov method) and confirmed numerically (Lyapunov exponents). In particular, it is shown that ϕ0 is effective at suppressing the chaotic behavior existing when ϕ0 = 0 over a wide range of values of the wave packet width, while the remaining parameters are kept constant.

Exponential-Algebraic Maps and Chaos in 3D Autonomous Quadratic Systems

2015 ◽  
Vol 25 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Vasiliy Ye. Belozyorov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Quadratic Systems ◽  
Discrete Maps ◽  
Algebraic Maps ◽  
Discrete Map

For some 3D autonomous quadratic dynamical systems an explicit autonomous exponential-algebraic 1D map, generating chaos in mentioned systems, is designed. Examples of the systems, where chaos is generated by such discrete maps, are given. New results about an existence of chaotic dynamics in the quadratic 3D systems are also derived. Besides, for the Lanford system (it is 3D autonomous quadratic dynamical system) the value of some parameter at which the system shows increased chaotic behavior is indicated. This assertion is based on the construction for the Lanford system of 2D exponential-algebraic discrete map which possesses chaotic properties.

CHAOS IN A THREE-DIMENSIONAL CANCER MODEL

2010 ◽  
Vol 20 (01) ◽  
pp. 71-79 ◽  
Author(s):  
MEHMET ITIK ◽  
STEPHEN P. BANKS
Keyword(s):  
Immune System ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Dynamical Model ◽  
Parameter Range ◽  
Cancer Growth ◽  
Cancer Model ◽  
Lyapunov Dimension

In this study, we develop a new dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to chaotic behavior. We explain the biological relevance of our model and the ways in which it differs from the existing ones. We perform equilibria analysis, indicate the conditions where chaotic dynamics can be observed, and show rigorously the existence of chaos by calculating the Lyapunov exponents and the Lyapunov dimension of the system. Moreover, we demonstrate that Shilnikov's theorem is valid in the parameter range of interest.

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