Related papers: The Van der Pol Equation
The equation of the Van der Pol oscillator, being characterized by a dissipative term, is non-Lagrangian. Appending an additional degree of freedom we bring the equation in the frame of action principle and thus introduce a one-way coupled…
We study a network of 500 coupled modified van der Pol oscillators. The value of a parameter associated with each oscillator is drawn from a normal distribution, giving a heterogeneous network. For strong enough coupling the oscillators all…
This study focuses on extending the concept of weak signal enhancement from dynamical systems based on vibrational resonance of nonlinear systems, to non-smooth systems. A Van der Pol- Duffing oscillator with a one-sided barrier, subjected…
Van der Pol equation (in short, vdP) as well as many its non-symmetric generalizations (the so-called van der Pol-like oscillators (in short, vdPl)) serve as nodes in coupled networks modeling real-life phenomena. Symmetric properties of…
Oscillator networks found in social and biological systems are characterized by the presence of wide ranges of coupling strengths and complex organization. Yet robustness and synchronization of oscillations are found to emerge on…
In this paper we study the existence, uniqueness and asymptotic stability of the periodic solutions for a Lipschitz system with a small right hand side. Classical hypotheses in the periodic case of second Bogolyubov's theorem imply our…
The limit cycle of the van der Pol oscillator, $\ddot{x}+ \epsilon (x^2-1) \dot{x} + x =0$, is studied in the plane $(x,\dot{x})$ by applying the homotopy analysis method. A recursive set of formulas that approximate the amplitude and form…
In this work, exact solutions of the nonlinear cubic-quintic Duffing-van der Pol oscillator with variable coefficients are obtained. Two approaches have been applied, one based on the factorization method combined with the Field Method, and…
Controlled transitions between a hierarchy of n-scroll attractors are investigated in a nonlinear optoelectronic oscillator. Using the system's feedback strength as a control parameter, it is shown experimentally the transition from Van der…
We outline formal and physical similarities between the quantum dynamics of open systems, and the mesoscopic description of classical systems affected by weak noise. The main tool of our interest is the dissipative Wigner equation, that,…
We discover presence of a hitherto unexplored type of resonance in a parametrically excited Van der Pol oscillator. The oscillator also possesses a state of anti-resonance. In the weak non-linear limit, we explain how to practically get a…
In this set of lectures, we review briefly some of the recent developments in the study of the chaotic dynamics of nonlinear oscillators, particularly of damped and driven type. By taking a representative set of examples such as the…
Following the paper of K. Shimizu et al. (2011) we consider the Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using…
Two-stroke relaxation oscillations consist of two distinct phases per cycle - one slow and one fast - which distinguishes them from the well-known van der Pol-type 'four-stroke' relaxation oscillations. This type of oscillation can be found…
This paper investigates the global dynamics of the discontinuous limit case of an archetypal oscillator with constant excitation that exhibits a single equilibrium. For parameter regions in which this oscillator possesses two or three…
The phenomenon of intrinsic localization in discrete nonlinear extended systems, i.e. the (generic) existence of discrete breathers, is shown to be not restricted to periodic solutions but it also extends to more complex (chaotic) dynamical…
We address the distinction between dynamical and additive noise in time series analysis by making a joint evaluation of both the statistical continuity of the series and the statistical differentiability of the reconstructed measure. Low…
We numerically simulate the two-dimensional inertial flow with the van der Waals effect in a straight periodic channel around the Poiseuille and Couette stationary states. Even though the flow remains laminar macroscopically, we observe…
Path integrals play a crucial role in describing the dynamics of physical systems subject to classical or quantum noise. In fact, when correctly normalized, they express the probability of transition between two states of the system. In…
We describe an extension of the Taylor method for the numerical solution of ODEs that uses Pad\'e approximants to obtain extremely precise numerical results. The accuracy of the results is essentially limited only by the computer time and…