THE GENERALIZED TIME-DELAYED HÉNON MAP: BIFURCATIONS AND DYNAMICS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350045
Author(s):  
SHAKIR BILAL ◽  
RAMAKRISHNA RAMASWAMY
Keyword(s):  
Lyapunov Exponents ◽  
Parameter Space ◽  
Periodic Motion ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Nonlinearity Parameter ◽  
High Dimensions ◽  
Low Dimensions ◽  
Onset Of Chaos ◽  
Generalized Time

We analyze the bifurcations of a family of time-delayed Hénon maps of increasing dimension and determine the regions where the motion is attracted to different dynamical states. As a function of parameters that govern nonlinearity and dissipation, boundaries that confine asymptotic periodic motion are determined analytically, and we examine their dependence on the dimension d. For large d these boundaries converge. In low dimensions both the period-doubling and quasiperiodic routes to chaos coexist in the parameter space, but for high dimensions the latter predominates and prior to the onset of chaos, the systems exhibit multistability. When the nonlinearity parameter is varied, the dimension of chaotic attractors in the systems changes smoothly with increasing number of non-negative Lyapunov exponents.

VARIETY OF TYPES OF CRITICAL BEHAVIOR AND MULTISTABILITY IN PERIOD-DOUBLING SYSTEMS WITH UNIDIRECTIONAL COUPLING NEAR THE ONSET OF CHAOS

1993 ◽  
Vol 03 (01) ◽  
pp. 139-152 ◽  
Author(s):  
A.P. KUZNETSOV ◽  
S.P. KUZNETSOV ◽  
I.R. SATAEV
Keyword(s):  
Parameter Space ◽  
Critical Behavior ◽  
Period Doubling ◽  
Control Parameters ◽  
Multicritical Point ◽  
Scaling Properties ◽  
Unidirectional Coupling ◽  
Critical Indices ◽  
Onset Of Chaos ◽  
Behavior Types

The dynamics of two unidirectionally coupled period-doubling systems is investigated depending on three relevant parameters (control parameters of subsystems and coupling). There is a hierarchy of critical behavior types. Feigenbaum’s critical surfaces existing in the parameter space are bounded by tricritical lines and intersect along the bicritical line. These lines, in turn, intersect at a new multicritical point BT. Universality and scaling properties for all the critical situations are discussed, and the table of critical indices is given.

Coexistence of Subharmonic Resonant Modes Obeying a Period-Adding Rule

2018 ◽  
Vol 28 (10) ◽  
pp. 1830031 ◽  
Author(s):  
Tiago Kroetz ◽  
Jefferson S. E. Portela ◽  
Ricardo L. Viana
Keyword(s):  
Parameter Space ◽  
Period Doubling ◽  
Hierarchical Organization ◽  
Chaotic Attractors ◽  
Engineering Systems ◽  
Electronic Circuits ◽  
Numerical Evidence ◽  
Resonant Modes ◽  
Periodic Domains ◽  
Subharmonic Resonances

We report numerical evidence of a novel chain of subharmonic resonances organized in period-adding domains in the parameter space. Different periodic domains are mediated by a coexisting main subharmonic motion according to a hierarchical organization rule. We consider the driven bilinear oscillator — which is the simplest system to model a range of engineering systems going from cracked mechanical structures to switching electronic circuits — and show that its parameter space is marked by multistability involving subharmonic main modes, as well as period-adding secondary modes which, through a period-doubling route, lead to coexisting chaotic attractors.

A New Imprisoned Strange Attractor

2019 ◽  
Vol 29 (13) ◽  
pp. 1950181
Author(s):  
Fahimeh Nazarimehr ◽  
Viet-Thanh Pham ◽  
Karthikeyan Rajagopal ◽  
Fawaz E. Alsaadi ◽  
Tasawar Hayat ◽  
...  
Keyword(s):  
Bifurcation Diagram ◽  
Lyapunov Exponents ◽  
Chaotic System ◽  
Strange Attractor ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Spherical Coordinate ◽  
New Chaotic System ◽  
Route To Chaos ◽  
Dynamical Properties

This paper proposes a new chaotic system with a specific attractor which is bounded in a sphere. The system is offered in the spherical coordinate. Dynamical properties of the system are investigated in this paper. The system shows multistability, and all of its attractors are inside or on the surface of the specific sphere. Bifurcation diagram of the system displays an inverse period-doubling route to chaos. Lyapunov exponents of the system are studied to show its chaotic attractors and predict its bifurcation points.

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.

Fold-Pitchfork Bifurcation, Arnold Tongues and Multiple Chaotic Attractors in a Minimal Network of Three Sigmoidal Neurons

2018 ◽  
Vol 28 (10) ◽  
pp. 1850123 ◽  
Author(s):  
Yo Horikawa ◽  
Hiroyuki Kitajima ◽  
Haruna Matsushita
Keyword(s):  
Periodic Solutions ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Quasiperiodic Solution ◽  
Chaotic Attractors ◽  
Dimensional Parameter ◽  
Chaotic Neural Networks ◽  
Minimal Network ◽  
Arnold Tongues ◽  
Period Doubling Bifurcations

Bifurcations and chaos in a network of three identical sigmoidal neurons are examined. The network consists of a two-neuron oscillator of the Wilson–Cowan type and an additional third neuron, which has a simpler structure than chaotic neural networks in the previous studies. A codimension-two fold-pitchfork bifurcation connecting two periodic solutions exists, which is accompanied by the Neimark–Sacker bifurcation. A stable quasiperiodic solution is generated and Arnold’s tongues emanate from the locus of the Neimark–Sacker bifurcation in a two-dimensional parameter space. The merging, splitting and crossing of the Arnold tongues are observed. Further, multiple chaotic attractors are generated through cascades of period-doubling bifurcations of periodic solutions in the Arnold tongues. The chaotic attractors grow and are destroyed through crises. Transient chaos and crisis-induced intermittency due to the crises are also observed. These quasiperiodic solutions and chaotic attractors are robust to small asymmetry in the output function of neurons.

Dynamics and Stability of Partial Immersion Milling Operations

1999 ◽  
Author(s):  
M. X. Zhao ◽  
B. Balachandran ◽  
M. A. Davies ◽  
J. R. Pratt
Keyword(s):  
Hopf Bifurcation ◽  
Time Variation ◽  
Periodic Motion ◽  
Contact Time ◽  
Period Doubling ◽  
Experimental Investigations ◽  
Partial Immersion ◽  
Milling Operations ◽  
Wide Range ◽  
The Stability

Abstract In this paper, numerical and experimental investigations conducted into the dynamics and stability of partial immersion milling operations are presented. A mechanics based model is used for simulations of a wide range of milling operations and instabilities that arise due to regeneration and/or impact effects are studied. Poincaré sections are used to assess the stability of motions. The studies reveal that apart from Hopf bifurcation of a periodic motion, a period-doubling bifurcation of a periodic motion may also lead to chatter in partial immersion milling operations. Issues such as tooth contact time variation and structure of stability charts are also discussed.

CHAOTIC DYNAMICS OF A SIMPLE PARAMETRICALLY DRIVEN DISSIPATIVE CIRCUIT

2011 ◽  
Vol 21 (07) ◽  
pp. 1927-1933 ◽  
Author(s):  
P. PHILOMINATHAN ◽  
M. SANTHIAH ◽  
I. RAJA MOHAMED ◽  
K. MURALI ◽  
S. RAJASEKAR
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classic Period ◽  
Control Signal ◽  
Period Doubling Bifurcation ◽  
Chaotic Circuit ◽  
Route To Chaos ◽  
Parameter Values

We introduce a simple parametrically driven dissipative second-order chaotic circuit. In this circuit, one of the circuit parameters is varied by an external periodic control signal. Thus by tuning the parameter values of this circuit, classic period-doubling bifurcation route to chaos is found to occur. The experimentally observed phenomena is further validated through corresponding numerical simulation of the circuit equations. The periodic and chaotic dynamics of this model is further characterized by computing Lyapunov exponents.

scholarly journals The study of chaotic and regular regimes of the fractal oscillators FitzHugh-Nagumo

2018 ◽  
Vol 62 ◽  
pp. 02017 ◽  
Author(s):  
Olga Lipko ◽  
Roman Parovik
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponents ◽  
Nerve Impulse ◽  
Chaotic Attractors ◽  
Control Parameters ◽  
Schmidt Orthogonalization ◽  
Non Local ◽  
Explicit Finite Difference ◽  
Maximum Lyapunov Exponents ◽  
Orthogonalization Procedure

In this paper we study the conditions for the existence of chaotic and regular oscillatory regimes of the hereditary oscillator FitzHugh-Nagumo (FHN), a mathematical model for the propagation of a nerve impulse in a membrane. To achieve this goal, using the non-local explicit finite-difference scheme and Wolf’s algorithm with the Gram-Schmidt orthogonalization procedure and the spectra of the maximum Lyapunov exponents were also constructed depending on the values of the control parameters of the model of the FHN. The results of the calculations showed that there are spectra of maximum Lyapunov exponents both with positive values and with negative values. The results of the calculations were also confirmed with the help of oscillograms and phase trajectories, which indicates the possibility of the existence of both chaotic attractors and limit cycles.

scholarly journals SINGULAR ORBITS AND DYNAMICS AT INFINITY OF A CONJUGATE LORENZ-LIKE SYSTEM

2015 ◽  
Vol 20 (2) ◽  
pp. 148-167 ◽  
Author(s):  
Fengjie Geng ◽  
Xianyi Li
Keyword(s):  
Theoretical Analysis ◽  
Lyapunov Exponents ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Basic Properties ◽  
Homoclinic And Heteroclinic Orbits ◽  
Quadratic Nonlinearities ◽  
Singular Orbits

A conjugate Lorenz-like system which includes only two quadratic nonlinearities is proposed in this paper. Some basic properties of this system, such as the distribution of its equilibria and their stabilities, the Lyapunov exponents, the bifurcations are investigated by some numerical and theoretical analysis. The forming mechanisms of compound structures of its new chaotic attractors obtained by merging together two simple attractors after performing one mirror operation are also presented. Furthermore, some of its other complex dynamical behaviours, which include the existence of singularly degenerate heteroclinic cycles, the existence of homoclinic and heteroclinic orbits and the dynamics at infinity, etc, are formulated in detail. In the meantime, some problems deserving further investigations are presented.

scholarly journals The onset of chaos in vortex sheet flow

2002 ◽  
Vol 454 ◽  
pp. 47-69 ◽  
Author(s):  
ROBERT KRASNY ◽  
MONIKA NITSCHE
Keyword(s):  
Vortex Ring ◽  
Vortex Core ◽  
Axisymmetric Flow ◽  
Period Doubling ◽  
Vortex Sheet ◽  
Small Scale ◽  
Poincaré Section ◽  
Poincare Section ◽  
Sheet Flow ◽  
Onset Of Chaos

Regularized point-vortex simulations are presented for vortex sheet motion in planar and axisymmetric flow. The sheet forms a vortex pair in the planar case and a vortex ring in the axisymmetric case. Initially the sheet rolls up into a smooth spiral, but irregular small-scale features develop later in time: gaps and folds appear in the spiral core and a thin wake is shed behind the vortex ring. These features are due to the onset of chaos in the vortex sheet flow. Numerical evidence and qualitative theoretical arguments are presented to support this conclusion. Past the initial transient the flow enters a quasi-steady state in which the vortex core undergoes a small-amplitude oscillation about a steady mean. The oscillation is a time-dependent variation in the elliptic deformation of the core vorticity contours; it is nearly time-periodic, but over long times it exhibits period-doubling and transitions between rotation and nutation. A spectral analysis is performed to determine the fundamental oscillation frequency and this is used to construct a Poincaré section of the vortex sheet flow. The resulting section displays the generic features of a chaotic Hamiltonian system, resonance bands and a heteroclinic tangle, and these features are well-correlated with the irregular features in the shape of the vortex sheet. The Poincaré section also has KAM curves bounding regions of integrable dynamics in which the sheet rolls up smoothly. The chaos seen here is induced by a self-sustained oscillation in the vortex core rather than external forcing. Several well-known vortex models are cited to justify and interpret the results.

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