Consider the initial-boundary value problem of the linear wave equation wtt − wxx = 0 on an interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system, while the boundary condition at the right endpoint has cubic nonlinearity of a van der Pol type. We show that the interactions of these linear and nonlinear boundary conditions can cause chaos t...

Consider the equation "v dv du = (1 ? u 2)v + ? u describing the orbits of the forced van der Pol equation " u + (u 2 ? 1) _ u + u = ; " a small parameter, (1) in the phase plane. The authors show the existence of two exceptional solutions corresponding to two conjugate complex exceptional values of the parameter near = 1 which remain close to the slow curve v = ?1=(u + 1) both in the direction...

The Van der Pol Mathieu equation, combining self-excitation and parametric excitation, is analysed near and at 1 : 2 resonance, using the averaging method. We analytically prove the existence of stable and unstable periodic solutions near the parametric resonance frequency. Above a certain detuning threshold, quasiperiodic solutions arise with basic periods of order 1 and order 1/ε where ε is t...

In this paper, the van der Pol equation with a time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manif...

We study dynamics of the Duffing–Van der Pol driven oscillator. Periodic steady-state solutions of the corresponding equation are determined within the Krylov-Bogoliubov-Mitropolsky approach to yield dependence of amplitude on forcing frequency as an implicit function, referred to as resonance curve or amplitude profile. Equations for singular points of resonance curves are solved exactly. We i...

In this paper, we study dynamics in delayed van der Pol–Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation. Both multiple time scales and center manifold reduction methods are applied to obtain the normal forms near a double Hopf critical point. A comparison between these two methods is given to show their equivalence. Bifurcations are classified in a two...

Grundlegende Faktoren in der Wahrnehmung von Websites sind Inhalt, Usability und Ästhetik (Thielsch, 2008). Ziel dieses Beitrages ist es die Bewertung der drei Konstrukte durch Webnutzer selbst näher zu beleuchten. Es werden Ergebnisse aus vier Studien vorgestellt (nges = 1772): In drei Befragungen (nges = 1408) zeigen sich hohe Übereinstimmungen zwischen den Einschätzungen der Nutzer, wobei si...

We study synchronization of two dissipatively coupled Van der Pol oscillators in the quantum regime. Due to quantum noise strict frequency locking is absent and is replaced by a crossover from weak to strong frequency entrainment. We discuss the differences to the behavior of one quantum Van der Pol oscillator subject to an external drive. Moreover, we describe a possible experimental realizati...