Chaotic Dynamical Behavior of Coupled One-Dimensional Wave Equations

2021 ◽  
Vol 31 (06) ◽  
pp. 2150115
Author(s):  
Fei Wang ◽  
Junmin Wang ◽  
Zhaosheng Feng
Keyword(s):  
Numerical Simulations ◽  
Wave Equations ◽  
Dynamical Behavior ◽  
Chaotic Oscillation ◽  
Van Der Pol ◽  
Velocity Feedback ◽  
One Dimensional ◽  
Theoretical Results ◽  
Dimensional Wave

In this paper, we consider the chaotic oscillation of coupled one-dimensional wave equations. The symmetric nonlinearities of van der Pol type are proposed at the two boundary endpoints, which can cause the energy of the system to rise and fall within certain bounds. At the interconnected point of the wave equations, the energy is injected into the system through an anti-damping velocity feedback. We prove the existence of the snapback repeller when the parameters enter a certain regime, which causes the system to be chaotic. Numerical simulations are presented to illustrate our theoretical results.

Chaotic Behaviors of One-Dimensional Wave Equations with van der Pol Boundary Conditions Containing a Source Term

2020 ◽  
Vol 30 (15) ◽  
pp. 2050227
Author(s):  
Zhijing Chen ◽  
Yu Huang ◽  
Haiwei Sun ◽  
Tongyang Zhou
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Source Term ◽  
Technical Difficulty ◽  
Van Der Pol ◽  
One Dimensional ◽  
Dynamical Method ◽  
Functional Envelope ◽  
The Right ◽  
Dimensional Wave

For one-dimensional wave equations with the van der Pol boundary conditions, there have been several different ways in the literature to characterize the complexity of their solutions. However, if the right-end van der Pol boundary condition contains a source term, then a considerable technical difficulty arises as to how to describe the complexity of the system. In this paper, we take advantage of a topologically dynamical method to characterize the dynamical behaviors of the systems, including sensitivity, transitivity and Li–Yorke chaos. For this end, we consider a system [Formula: see text] induced by a sequence of continuous maps and its functional envelope [Formula: see text], and show that, under some considerable condition, [Formula: see text] is transitive if and only if [Formula: see text] is weakly mixing of order [Formula: see text]; [Formula: see text] is Li–Yorke chaotic and sensitive if [Formula: see text] is strongly mixing. Those abstract results have their own significance and can be applied to such kind of equations.

On the metric perturbations of the Reissner–Nordström black hole

1979 ◽  
Vol 367 (1728) ◽  
pp. 1-14 ◽  
Keyword(s):  
Black Hole ◽  
Wave Equations ◽  
Schwarzschild Black Hole ◽  
One Dimensional ◽  
Dimensional Wave

The two pairs of one-dimensional wave equations which govern the odd and the even-parity perturbations of the Reissner–Nordström black hole are derived directly from a treatment of its metric perturbations. The treatment closely parallels the corresponding treatment in the context of the Schwarzschild black hole.

DYNAMICS OF A PREY-DEPENDENT DIGESTIVE MODEL WITH STATE-DEPENDENT IMPULSIVE CONTROL

2012 ◽  
Vol 22 (04) ◽  
pp. 1250092 ◽  
Author(s):  
LINNING QIAN ◽  
QISHAO LU ◽  
JIARU BAI ◽  
ZHAOSHENG FENG
Keyword(s):  
Numerical Simulations ◽  
Analytical Method ◽  
Impulsive Control ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Lambert W Function ◽  
Impulsive Effect ◽  
State Dependent ◽  
Theoretical Results

In this paper, we study the dynamical behavior of a prey-dependent digestive model with a state-dependent impulsive effect. Using the Poincaré map and the Lambert W-function, we find the analytical expression of discrete mapping. Sufficient conditions are established for transcritical bifurcation and period-doubling bifurcation through an analytical method. Exact locations of these bifurcations are explored. Numerical simulations of an example are illustrated which agree well with our theoretical results.

scholarly journals Parabolic bursting, spike-adding, dips and slices in a minimal model

2019 ◽  
Vol 14 (4) ◽  
pp. 406
Author(s):  
Mathieu Desroches ◽  
Jean-Pierre Francoise ◽  
Martin Krupa
Keyword(s):  
Qualitative Analysis ◽  
Numerical Simulations ◽  
Phase Portrait ◽  
Minimal Model ◽  
Bifurcation Theory ◽  
Minimal System ◽  
Dimensional System ◽  
Slow Flow ◽  
Van Der Pol ◽  
One Dimensional

A minimal system for parabolic bursting, whose associated slow flow is integrable, is presented and studied both from the viewpoint of bifurcation theory of slow-fast systems, of the qualitative analysis of its phase portrait and of numerical simulations. We focus the analysis on the spike-adding phenomenon. After a reduction to a periodically forced one-dimensional system, we uncover the link with the dips and slices first discussed by J.E. Littlewood in his famous articles on the periodically forced van der Pol system.

scholarly journals ANALYSIS OF TORUS BREAKDOWN INTO CHAOS IN A CONSTRAINT DUFFING VAN DER POL OSCILLATOR

2008 ◽  
Vol 18 (04) ◽  
pp. 1051-1068 ◽  
Author(s):  
MUNEHISA SEKIKAWA ◽  
NAOHIKO INABA ◽  
TAKASHI TSUBOUCHI ◽  
KAZUYUKI AIHARA
Keyword(s):  
Bifurcation Diagram ◽  
Van Der Pol Oscillator ◽  
Circle Map ◽  
Bifurcation Structure ◽  
Van Der Pol ◽  
One Dimensional ◽  
Implicit Equations ◽  
Periodic State ◽  
Torus Breakdown ◽  
Theoretical Results

The bifurcation structure of a constraint Duffing van der Pol oscillator with a diode is analyzed and an objective bifurcation diagram is illustrated in detail in this work. An idealized case, where the diode is assumed to operate as a switch, is considered.In this case, the Poincaré map is constructed as a one-dimensional map: a circle map. The parameter boundary between a torus-generating region where the circle map is a diffeomorphism and a chaos-generating region where the circle map has extrema is derived explicitly, without solving the implicit equations, by adopting some novel ideas. On the bifurcation diagram, intermittency and a saddle-node bifurcation from the periodic state to the quasi-periodic state can be exactly distinguished. Laboratory experiment is also carried out and theoretical results are verified.

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