ON THE BIFURCATION PHENOMENA OF THE KURAMOTO–SIVASHINSKY EQUATION

1993 ◽  
Vol 03 (05) ◽  
pp. 1299-1303 ◽  
Author(s):  
FRED FEUDEL ◽  
ULRIKE FEUDEL ◽  
AXEL BRANDENBURG
Keyword(s):  
Periodic Solution ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Transient Chaos ◽  
Bifurcation Parameter ◽  
Periodic Motions ◽  
Chaotic Region ◽  
Bifurcation Phenomena ◽  
Route To Chaos ◽  
Sivashinsky Equation

Bifurcation phenomena of the Kuramoto–Sivashinsky equation have been studied numerically. The solutions considered are restricted to the invariant subspace of odd functions. One possible route to chaos via a period-doubling cascade is investigated in detail: The four-modal steady-state loses its stability through a Hopf bifurcation and a branch of periodic motions is created. After a symmetry breaking the periodic solution undergoes a period-doubling cascade which ends up in two antisymmetric chaotic attractors. A merging of these antisymmetric attractors to a symmetric one is observed. The chaotic branch depending on the bifurcation parameter is characterized by the values of the Lyapunov exponents. Periodic windows within the chaotic region are also detected. Finally, a further increase of the bifurcation parameter leads to a transition from the attractor into transient chaos.

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.

Bifurcation Analysis of a Belousov–Zhabotinsky Reaction Model

2015 ◽  
Vol 25 (06) ◽  
pp. 1550093 ◽  
Author(s):  
Xiaoli Wang ◽  
Yu Chang ◽  
Dashun Xu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Reaction Model ◽  
Bifurcation Phenomena ◽  
Route To Chaos ◽  
The Stability ◽  
Belousov Zhabotinsky Reaction

We investigate the bifurcation phenomena in a Belousov–Zhabotinsky reaction model by applying Hopf bifurcation theory in frequency domain and harmonic balance method. The high accurate predictions, i.e. fourth-order harmonic balance approximation, on frequencies, amplitudes, and approximation expressions for periodic solutions emerging from Hopf bifurcation are provided. We also detect the stability and location of these periodic solutions. Numerical simulations not only confirm the theoretical analysis results but also illustrate some complex oscillations such as a cascade of period-doubling bifurcation, quasi-periodic solution, and period-doubling route to chaos. All these results improve the understanding of the dynamics of the model.

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.

Controlling Nonlinear Dynamics in a Two-Well Impact System.

1998 ◽  
Vol 08 (12) ◽  
pp. 2387-2407 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Nonlinear Dynamics ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
System Response ◽  
Chaotic Attractors ◽  
Impact System ◽  
Local Bifurcations ◽  
Global Optimal ◽  
Route To Chaos

A method for controlling nonlinear dynamics based on avoiding homoclinic intersection is systematically implemented to perform a numerical analysis of the control induced modifications of the steady attractors and bifurcation scenario of a two-well impact system. The work is divided into two parts. This paper (Part I) deals with the analysis of the harmonic (reference) and global optimal excitations, which are both symmetric. The bifurcation diagrams obtained for increasing values of the excitation amplitude show there exist a "basic" attractor and other "complementary" solutions. The range of stability of the principal complementary attractors is numerically established, and the mechanisms leading to their disapperance are identified. The role of classical and nonclassical local bifurcations in determining the system response is emphasized. Chaotic attractors are seen to appear and disappear both by classical period doubling route to chaos and by sudden changes. Subductions, boundary and interior crises are repeatedly observed. By comparison of the system response under different excitations we obtain information on the performances of global control, which furnishes relatively low gain in terms of regularization but succeeds in controlling the whole phase space.

CLASSIFICATION OF BIFURCATIONS AND CHAOS IN CHUA'S CIRCUIT WITH EFFECT OF DIFFERENT PERIODIC FORCES

2009 ◽  
Vol 19 (06) ◽  
pp. 1951-1973 ◽  
Author(s):  
K. SRINIVASAN ◽  
K. THAMILMARAN ◽  
A. VENKATESAN
Keyword(s):  
Period Doubling ◽  
Phase Portraits ◽  
Chua’S Circuit ◽  
Structure Generation ◽  
Chua's Circuit ◽  
Bifurcation Phenomena ◽  
Route To Chaos ◽  
Doubling Sequence ◽  
Two Parameter

We study the effect of different periodic excitations like sine, square, triangle and sawtooth waves on Chua's circuit and show that the circuit can undergo distinctly modified bifurcation structure, generation of new periodic regimes, induction of crises and so on. In particular, we point out that under the influence of different periodic excitations, a rich variety of bifurcation phenomena, including the familiar period-doubling sequence, intermittent route to chaos and period-adding sequences, reverse bifurcations, remerging chaotic band attractors, a large number of coexisting periodic attractors exist in the system. The analysis is carried out numerically using phase portraits, two-parameter phase diagrams in the forcing amplitude-frequency plane and one-parameter bifurcation diagrams. The chaotic dynamics of this circuit is also realized experimentally.

scholarly journals A Route to Chaos in the Boros–Moll Map

2019 ◽  
Vol 29 (04) ◽  
pp. 1930009 ◽  
Author(s):  
Laura Gardini ◽  
Víctor Mañosa ◽  
Iryna Sushko
Keyword(s):  
Specific Property ◽  
Parameter Family ◽  
Peculiar Feature ◽  
Chaotic Region ◽  
Bifurcation Phenomena ◽  
The Family ◽  
Homoclinic Bifurcations ◽  
Route To Chaos ◽  
Special Case ◽  
Critical Lines

The Boros–Moll map appears as a subsystem of a Landen transformation associated to certain rational integrals and its dynamics is related to their convergence. In the paper, we study the dynamics of a one-parameter family of maps which unfold the Boros–Moll one, showing that the existence of an unbounded invariant chaotic region in the Boros–Moll map is a peculiar feature within the family. We relate this singularity with a specific property of the critical lines that occurs only for this special case. In particular, we explain how the unbounded chaotic region in the Boros–Moll map appears. We especially explain the main contact/homoclinic bifurcations that occur in the family. We also report some other bifurcation phenomena that appear in the considered unfolding.

A Dream that has Come True: Chaos from a Nonlinear Circuit with a Real Memristor

2020 ◽  
Vol 30 (13) ◽  
pp. 2030036
Author(s):  
Christos K. Volos ◽  
Viet-Thanh Pham ◽  
Hector E. Nistazakis ◽  
Ioannis N. Stouboulos
Keyword(s):  
Chaos Theory ◽  
Period Doubling ◽  
Nonlinear Circuits ◽  
Nonlinear Element ◽  
Chaotic Attractors ◽  
Nonlinear Circuit ◽  
Coexisting Attractors ◽  
Circuit Element ◽  
Route To Chaos ◽  
First Time

In the last decade, researchers, who work in the field of nonlinear circuits, have the “dream” to use a real memristor, which is the only nonlinear fundamental circuit element, in a new or other reported nonlinear circuit in literature, in order to experimentally investigate chaos. With this intention, for the first time, a well-known nonlinear circuit, in which its nonlinear element has been replaced with a commercially available memristor (KNOWM memristor), is presented in this work. Interesting phenomena concerning chaos theory, such as period-doubling route to chaos, coexisting attractors, one-scroll and double-scroll chaotic attractors are experimentally observed.

scholarly journals Complex dynamics and coexistence of period-doubling and period-halving bifurcations in an integrated pest management model with nonlinear impulsive control

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Changtong Li ◽  
Sanyi Tang ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Pest Control ◽  
Impulsive Control ◽  
Period Doubling ◽  
Dynamical Behaviour ◽  
Period Doubling Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model

Abstract An expectation for optimal integrated pest management is that the instantaneous numbers of natural enemies released should depend on the densities of both pest and natural enemy in the field. For this, a generalised predator–prey model with nonlinear impulsive control tactics is proposed and its dynamics is investigated. The threshold conditions for the global stability of the pest-free periodic solution are obtained based on the Floquet theorem and analytic methods. Also, the sufficient conditions for permanence are given. Additionally, the problem of finding a nontrivial periodic solution is confirmed by showing the existence of a nontrivial fixed point of the model’s stroboscopic map determined by a time snapshot equal to the common impulsive period. In order to address the effects of nonlinear pulse control on the dynamics and success of pest control, a predator–prey model incorporating the Holling type II functional response function as an example is investigated. Finally, numerical simulations show that the proposed model has very complex dynamical behaviour, including period-doubling bifurcation, chaotic solutions, chaos crisis, period-halving bifurcations and periodic windows. Moreover, there exists an interesting phenomenon whereby period-doubling bifurcation and period-halving bifurcation always coexist when nonlinear impulsive controls are adopted, which makes the dynamical behaviour of the model more complicated, resulting in difficulties when designing successful pest control strategies.

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

DEFECT-LINE DYNAMICS IN OSCILLATORY SPIRAL WAVES

2006 ◽  
Vol 06 (04) ◽  
pp. L379-L386
Author(s):  
STEVEN WU
Keyword(s):  
Critical Point ◽  
Period Doubling ◽  
Spiral Wave ◽  
Chaotic Regime ◽  
Spiral Waves ◽  
Bifurcation Parameter ◽  
Local Dynamics ◽  
Total Length ◽  
Defect Line ◽  
Large Fluctuations

We study defect-line dynamics in a 2-D spiral-wave pair in the Rössler model for its underlying local dynamics in period-N and chaotic regimes with a single bifurcation parameter κ. We find that a spiral wave pair is always stable across the period-doubling cascade and in the chaotic regime. When N ≥ 2 defect lines appear spontaneously and a loop exchange occurs across the defect line. There exists a "critical point" κ c below and above which the time-averaged total length of defect lines L converges to almost constant but different values L1 and L2. When κ > κ c defect lines show large fluctuations due to creation and annihilation processes.

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