Basins of attraction for chaotic attractors in coupled systems with period doubling

1997 ◽  
Vol 23 (2) ◽  
pp. 144-146
Author(s):  
B. P. Bezruchko ◽  
E. P. Seleznev
Keyword(s):  
Period Doubling ◽  
Basins Of Attraction ◽  
Coupled Systems ◽  
Chaotic Attractors

Antimonotonicity, Chaos and Multiple Attractors in a Novel Autonomous Jerk Circuit

2017 ◽  
Vol 27 (07) ◽  
pp. 1750100 ◽  
Author(s):  
J. Kengne ◽  
A. Nguomkam Negou ◽  
Z. T. Njitacke
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Semiconductor Diode ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Multiple Attractors ◽  
Systematic Analysis ◽  
The Stability

We perform a systematic analysis of a system consisting of a novel jerk circuit obtained by replacing the single semiconductor diode of the original jerk circuit described in [Sprott, 2011a] with a pair of semiconductor diodes connected in antiparallel. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a control system with nonlinear velocity feedback. The stability of the (unique) fixed point, the local bifurcations, and the discrete symmetries of the model equations are discussed. The complex behavior of the system is categorized in terms of its parameters by using bifurcation diagrams, Lyapunov exponents, time series, Poincaré sections, and basins of attraction. Antimonotonicity, period doubling bifurcation, symmetry restoring crises, chaos, and coexisting bifurcations are reported. More interestingly, one of the key contributions of this work is the finding of various regions in the parameters’ space in which the proposed (“elegant”) jerk circuit experiences the unusual phenomenon of multiple competing attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). The basins of attraction of various coexisting attractors display complexity (i.e. fractal basins boundaries), thus suggesting possible jumps between coexisting attractors in experiment. Results of theoretical analyses are perfectly traced by laboratory experimental measurements. To the best of the authors’ knowledge, the jerk circuit/system introduced in this work represents the simplest electrical circuit (only a quadruple op amplifier chip without any analog multiplier chip) reported to date capable of four disconnected periodic and chaotic attractors for the same parameters setting.

CHARACTERIZATION OF CHAOTIC ATTRACTORS INSIDE BAND-MERGING SCENARIO IN A ZAD-CONTROLLED BUCK CONVERTER

2012 ◽  
Vol 22 (10) ◽  
pp. 1230034
Author(s):  
JOHN ALEXANDER TABORDA ◽  
FABIOLA ANGULO ◽  
GERARD OLIVAR
Keyword(s):  
Control Strategy ◽  
Control Parameter ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Buck Converter ◽  
Chaotic Attractors ◽  
Complex Dynamic ◽  
Self Similar ◽  
Dynamic Links

Zero Average Dynamics (ZAD) control strategy has been developed, applied and widely analyzed in the last decade. Numerous and interesting phenomena have been studied in systems controlled by ZAD strategy. In particular, the ZAD-controlled buck converter has been a source of nonlinear and nonsmooth phenomena, such as period-doubling, merging bands, period-doubling bands, torus destruction, fractal basins of attraction or codimension-2 bifurcations. In this paper, we report a new bifurcation scenario found inside band-merging scenario of ZAD-controlled buck converter. We use a novel qualitative framework named Dynamic Linkcounter (DLC) approach to characterize chaotic attractors between consecutive crisis bifurcations. This approach complements the results that can be obtained with Bandcounter approaches. Self-similar substructures denoted as Complex Dynamic Links (CDLs) are distinguished in multiband chaotic attractors. Geometrical changes in multiband chaotic attractors are detected when the control parameter of ZAD strategy is varied between two consecutive crisis bifurcations. Linkcount subtracting staircases are defined inside band-merging scenario.

scholarly journals Sequence of Routes to Chaos in a Lorenz-Type System

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Fangyan Yang ◽  
Yongming Cao ◽  
Lijuan Chen ◽  
Qingdu Li
Keyword(s):  
Limit Cycles ◽  
Chaotic Attractor ◽  
Type System ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Main Route ◽  
Route To Chaos ◽  
Period Doubling Bifurcations

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

scholarly journals Stochastic sensitivity of quasiperiodic and chaotic attractors of the discrete Lotka-Volterra model

2020 ◽  
Vol 55 ◽  
pp. 19-32
Author(s):  
A.V. Belyaev ◽  
T.V. Perevalova
Keyword(s):  
Period Doubling ◽  
Basins Of Attraction ◽  
Invariant Curve ◽  
Volterra Model ◽  
Chaotic Attractors ◽  
Deterministic System ◽  
System A ◽  
Stochastic Sensitivity ◽  
Two Parameters ◽  
Random States

The aim of the study presented in this article is to analyze the possible dynamic modes of the deterministic and stochastic Lotka-Volterra model. Depending on the two parameters of the system, a map of regimes is constructed. Parametric areas of existence of stable equilibria, cycles, closed invariant curves, and also chaotic attractors are studied. The bifurcations such as the period doubling, Neimark-Sacker and the crisis are described. The complex shape of the basins of attraction of irregular attractors (closed invariant curve and chaos) is demonstrated. In addition to the deterministic system, the stochastic system, which describes the influence of external random influence, is discussed. Here, the key is to find the sensitivity of such complex attractors as a closed invariant curve and chaos. In the case of chaos, an algorithm to find critical lines giving the boundary of a chaotic attractor, is described. Based on the found function of stochastic sensitivity, confidence domains are constructed that allow us to describe the form of random states around a deterministic attractor.

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.

THE EFFECT OF NONLINEAR DAMPING ON THE UNIVERSAL ESCAPE OSCILLATOR

1999 ◽  
Vol 09 (04) ◽  
pp. 735-744 ◽  
Author(s):  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Energy Dissipation ◽  
Period Doubling ◽  
Nonlinear Damping ◽  
Basins Of Attraction ◽  
Period Doubling Bifurcation ◽  
Damping Coefficient ◽  
Nonlinear Dissipation ◽  
Fractal Basin Boundaries ◽  
Basin Boundaries

This paper analyzes the role of nonlinear dissipation on the universal escape oscillator. Nonlinear damping terms proportional to the power of the velocity are assumed and an investigation on its effects on the dynamics of the oscillator, such as the threshold of period-doubling bifurcation, fractal basin boundaries and how the basins of attraction are destroyed, is carried out. The results suggest that increasing the power of the nonlinear damping, has similar effects as of decreasing the damping coefficient for a linearly damped case, showing the very importance of the level or amount of energy dissipation.

Fractal Basin Boundaries and Chaotic Motions in Spring-Pendulum System

2003 ◽  
Author(s):  
Aria Alasty ◽  
Rasool Shabani
Keyword(s):  
Numerical Simulations ◽  
Multiple Scales ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Chaotic Motions ◽  
Pendulum System ◽  
Fractal Basin Boundaries ◽  
System Parameters ◽  
Basin Boundaries

This study investigates chaotic response in the spring-pendulum system. In this system beside of strange attractors, multiple regular attractors may coexist for some values of system parameters, where it is important to study the global behavior of the system using the basin boundaries of the attractors. Multiple scales method is used to distinguish the regions of stable and unstable attractors. In unstable regions, bifurcation diagram and poincare´ maps are used to study the existence of quasi-periodic and chaotic attractors. Results show that the jumping phenomena may occur when multiple regular attractors exist and for this case fractal basins of attraction are developed using numerical simulations.

scholarly journals A Dynamic Duopoly Model: When a Firm Shares the Market with Certain Profit

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1826
Author(s):  
Sameh S. Askar
Keyword(s):  
Equilibrium Point ◽  
Market Share ◽  
Period Doubling ◽  
Analytical Investigation ◽  
Basins Of Attraction ◽  
Stability Conditions ◽  
Cournot Duopoly ◽  
Demand Functions ◽  
Duopoly Model ◽  
Dynamic Duopoly

The current paper analyzes a competition of the Cournot duopoly game whose players (firms) are heterogeneous in a market with isoelastic demand functions and linear costs. The first firm adopts a rationally-based gradient mechanism while the second one chooses to share the market with certain profit in order to update its production. It trades off between profit and market share maximization. The equilibrium point of the proposed game is calculated and its stability conditions are investigated. Our studies show that the equilibrium point becomes unstable through period doubling and Neimark–Sacker bifurcation. Furthermore, the map describing the proposed game is nonlinear and noninvertible which lead to several stable attractors. As in literature, we have provided an analytical investigation of the map’s basins of attraction that includes lobes regions.

CHAOTIC BEHAVIOR INDUCED BY THERMAL MODULATION IN A MODEL OF THE CONVECTIVE FLOW OF A FLUID MIXTURE IN A POROUS MEDIUM

1999 ◽  
Vol 09 (02) ◽  
pp. 383-396 ◽  
Author(s):  
J.-M. MALASOMA ◽  
P. WERNY ◽  
C.-H. LAMARQUE
Keyword(s):  
Porous Medium ◽  
Convective Flow ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Type I ◽  
Fluid Mixture ◽  
Numerical Investigations ◽  
Period Doubling Bifurcations

Numerical investigations of the global behavior of a model of the convective flow of a binary mixture in a porous medium are reported. We find a complex behavior characterized by the presence of coexisting periodic, quasiperiodic and chaotic attractors. Bifurcations of periodic solutions and routes to chaos via type-I intermittency and period-doubling bifurcations are described. Boundary crises and band merging crises have also been observed.

CHAOTIC SYNCHRONIZATION AND ANTISYNCHRONIZATION IN COUPLED SINE MAPS

2005 ◽  
Vol 15 (07) ◽  
pp. 2161-2177 ◽  
Author(s):  
V. L. MAISTRENKO ◽  
YU. L. MAISTRENKO ◽  
E. MOSEKILDE
Keyword(s):  
Parameter Space ◽  
Period Doubling ◽  
Chaotic Synchronization ◽  
Basins Of Attraction ◽  
Main Diagonal ◽  
Different Types ◽  
Regions Of Stability ◽  
The Individual ◽  
Period Doubling Bifurcations

This paper investigates different types of chaotic synchronization in a system of two coupled sine maps. Due to the bimodal nature of the individual map, there is a range of parameters in which two synchronized chaotic states coexist along the main diagonal. In certain parameter regions, various (regular or chaotic) asynchronous states coexist with the synchronized chaotic states, and the basins of attraction become quite complicated. We determine the regions of stability for the so-called principal cycles that arise through transverse period-doubling bifurcations of synchronized saddle cycles. Particular emphasis is paid to the occurrence of chaotic antisynchronization, the coexistence of antisynchronous chaotic states, and the presence of narrow regions of parameter space in which states of chaotic synchronization and antisynchronization exist simultaneously. For each of these cases we provide detailed pictures of the associated basin structures.

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