Chaotic behaviors of one dimensional wave equations with van der Pol nonlinear boundary conditions

2018 ◽  
Vol 59 (2) ◽  
pp. 022704 ◽  
Author(s):  
Zhijing Chen ◽  
Tingwen Huang ◽  
Yu Huang ◽  
Xin Liu
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Van Der Pol ◽  
One Dimensional ◽  
Dimensional Wave

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.

scholarly journals Continuum of one-sign solutions of one-dimensional Minkowski-curvature problem with nonlinear boundary conditions

2021 ◽  
Author(s):  
Yanqiong Lu ◽  
Zhengqi Jing
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Global Bifurcation ◽  
Nonlinear Boundary Conditions ◽  
Continuous Functions ◽  
One Dimensional ◽  
Curvature Equation ◽  
Existence And Multiplicity ◽  
Curvature Problem ◽  
The Continuum

In this work, we investigate the continuum of one-sign solutions of the nonlinear one-dimensional Minkowski-curvature equation $$-\big(u’/\sqrt{1-\kappa u’^2}\big)’=\lambda f(t,u),\ \ t\in(0,1)$$ with nonlinear boundary conditions $u(0)=\lambda g_1(u(0)), u(1)=\lambda g_2(u(1))$ by using unilateral global bifurcation techniques, where $\kappa>0$ is a constant, $\lambda>0$ is a parameter $g_1,g_2:[0,\infty)\to (0,\infty)$ are continuous functions and $f:[0,1]\times[-\frac{1}{\sqrt{\kappa}},\frac{1}{\sqrt{\kappa}}]\to\mathbb{R}$ is a continuous function. We prove the existence and multiplicity of one-sign solutions according to different asymptotic behaviors of nonlinearity near zero.

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.

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