Related papers: The Van der Pol Equation
We report a transition from homogeneous steady state to inhomogeneous steady state in coupled oscillators, both limit cycle and chaotic, under cyclic coupling and diffusive coupling as well when an asymmetry is introduced in terms of a…
This study introduces the reader to the theory of approximating the solution(s) of a non-linear, second order, ordinary differential equation (ODE) with piecewise polynomial functions by using the collocation method. It then focuses on the…
Wave propagation and acoustic scattering problems require vast computational resources to be solved accurately at high frequencies. Asymptotic methods can make this cost potentially frequency independent by explicitly extracting the…
An attractor of a piecewise-smooth continuous system of differential equations can bifurcate from a stable equilibrium to a more complicated invariant set when it collides with a switching manifold under parameter variation. Here numerical…
In this paper, we characterize a class of solutions to the unsteady 2-dimensional flow of a van der Waals fluid involving shock waves, and derive an asymptotic amplitude equation exhibiting quadratic and cubic nonlinearities including…
Chaos in classical systems has been studied in plenty over many years. Although the search for chaos in quantum systems has been an area of prominent research over the last few decades, the detailed analysis of many inherently chaotic…
By using a generalization of the multiple scales technique we develop a method to derive amplitude equations for zero--dimensional forced systems. The method allows to consider either additive or multiplicative forcing terms and can be…
The new phenomenological model of the ensemble of three neurons with chemical (synaptic) and electrical couplings has been studied. One neuron is modeled by a single Van der Pol oscillator. The influence of the electrical coupling strength…
This article is concerned in establishing the existence and regularity of solution of semi-hyperbolic patch problem for two-dimensional isentropic Euler equations with van der Waals gas. This type of solution appears in the transonic flow…
The purpose of this work is twofold. First, we construct probabilistically strong solutions to the three-dimensional Euler equations perturbed by additive noise that are $\mathbb{P}$-almost surely continuous in time, H\"older in space, and…
A method is described for predicting statistical properties of turbulence. Collections of Fourier amplitudes are represented by nonuniformly spaced modes with enhanced coupling coefficients. The statistics of the full dynamics can be…
Astrophysical objects frequently exhibit some irregularities or complex behaviour in their light curves. We focus primarily on hot stars, where both radial and non-radial pulsations are observed. One of the primary research goals is to…
Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to demonstrate these…
We discuss the notion of resonance, as well as the existence and uniqueness of periodic solutions for a forced simple harmonic oscillator. While this topic is elementary, and well-studied for sinusoidal forcing, this does not seem to be the…
We construct an analytical theory of interplay between synchronizing effects by common noise and by global coupling for a general class of smooth limit-cycle oscillators. Both the cases of attractive and repulsive coupling are considered.…
In this paper, the theory of harmonic maps is extended. The soliton or traveling wave solutions of Euler's equations of the extended harmonic maps are studied. In certain cases, the chaotic behaviors of these partial equations can be found…
In the absence of external forcing, all trajectories on the phase plane of the van der Pol oscillator tend to a closed, periodic, trajectory -- the limit cycle -- after infinite time. Here, we drive the van der Pol oscillator with an…
We connect quantum graphs with infinite leads, and turn them to scattering systems. We show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics: typical poles, delay…
The paper introduces a new 3D strange attractor topologically different from any other known chaotic attractors. The intentionally constructed model of three autonomous first-order differential equations derives from the coupling-induced…
In this tutorial, three examples of stochastic systems are considered: A strongly-damped oscillator, a weakly-damped oscillator and an undamped oscillator (integrator) driven by noise. The evolution of these systems is characterized by the…