Forced Van Der Pol Equation Research Articles

Effects of amplitude modulation on mixed-mode oscillations in the forced van der Pol equation

Effects of amplitude modulation on mixed-mode oscillations in the forced van der Pol equation

Resonances of a forced van der Pol equation with parametric damping

Resonances of a forced van der Pol equation with parametric damping

ANALYTICAL SOLUTION OF MODIFIED FORCED VAN DER POL VIBRATION EQUATION USING MODIFIED HARMONIC BALANCE METHOD

For obtaining analytical solutions to the modified forced Van der Pol equation, the modified harmonic balance method has been developed. In classical harmonic balance method, the numerical procedure is used for solving a set of nonlinear algebraic equations. But it requires laborious computational attempt and accurate primary guesses values which makes it very hard-working to calculate. According to our method, a set of nonlinear algebraic equations has been converted to a set of linear algebraic equations by using a nonlinear one instead of a set. As a result, it reduces the massive computational work. This method provides not only better results than the existing harmonic balance method but also provides very close solutions to the corresponding numerical results. It is noticeable that there is substantial similarity between the approximate and the numerical results attained by the fourth order Runge-Kutta method. Moreover, the method is facile and straightforward. This technique may play great role to handle strongly nonlinear damped systems with external forces.

Controlling Chaos—Forced van der Pol Equation

Nonlinear systems are typically linearized to permit linear feedback control design, but, in some systems, the nonlinearities are so strong that their performance is called chaotic, and linear control designs can be rendered ineffective. One famous example is the van der Pol equation of oscillatory circuits. This study investigates the control design for the forced van der Pol equation using simulations of various control designs for iterated initial conditions. The results of the study highlight that even optimal linear, time-invariant (LTI) control is unable to control the nonlinear van der Pol equation, but idealized nonlinear feedforward control performs quite well after an initial transient effect of the initial conditions. Perhaps the greatest strength of ideal nonlinear control is shown to be the simplicity of analysis. Merely equate coefficients order-of-differentiation insures trajectory tracking in steady-state (following dissipation of transient effects of initial conditions), meanwhile the solution of the time-invariant linear-quadratic optimal control problem with infinite time horizon is needed to reveal constant control gains for a linear-quadratic regulator. Since analytical development is so easy for ideal nonlinear control, this article focuses on numerical demonstrations of trajectory tracking error.

From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation

In this article, we study canard solutions of the forced van der Pol equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation made herein is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. In the low-frequency forcing regime, we demonstrate the existence of primary maximal canards induced by folded saddle nodes of type I and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. Then, we turn to the intermediate- and high-frequency forcing regimes and show that the forced van der Pol possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms the central numerical observation that the folds of the maximal canards created in the low-frequency regime continue directly into the folds of the maximal torus canards that exist in the intermediate- and high-frequency regimes. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that the fold curves of the secondary canards turn around in the intermediate-frequency regime, instead of continuing into the high-frequency regime. Also, we identify the mechanism responsible for this turning. Finally, we show that the forced van der Pol equation is a normal form-type equation for a class of single-frequency periodically driven slow/fast systems with two fast variables and one slow variable which possess a non-degenerate fold of limit cycles. The analytic techniques used herein rely on geometric desingularisation, invariant manifold theory, Melnikov theory, and normal form methods. The numerical methods used herein were developed in Desroches et al. (SIAM J Appl Dyn Syst 7:1131–1162, 2008, Nonlinearity 23:739–765 2010).

Numerical studies of a plasma diode with external forcing

With reference to laboratory Q-machine studies we analyze the dynamics of a plasma diode under external forcing. Assuming a strong axial magnetic field, the problem is analyzed in one spatial dimension by a particle-in-cell code. The cathode is assumed to be operated in electron rich conditions, supplying an abundance of electrons. We compare different forcing schemes with the results obtained by solving the van der Pol equation. In one method of forcing we apply an oscillation in addition to the DC end plate bias and consider both amplitude and frequency variations. An alternative method of perturbation consists of modelling an absorbing grid at some internal position. Also in this case we can have a constant frequency with varying amplitude or alternatively an oscillation with chirped frequency but constant amplitude. We find that the overall features of the forced van der Pol equation are recovered, but the details in the plasma response need more attention to the harmonic responses, requiring extensions of the model equation. The analysis is extended by introducing collisional effects, where we emphasize charge exchange collisions of ions, since these processes usually have the largest cross sections and give significant modifications of the diode performance. In particular we find a reduction in oscillator frequency, although a linear scaling of the oscillation time with the system length remains also in this case.

Computation of symbolic dynamics for one-dimensional maps

In this paper we design and implement rigorous algorithms for computing symbolic dynamics for piecewise-monotone-continuous maps of the interval. The algorithms are based on computing forwards and backwards approximations of the boundary, discontinuity and critical points. We explain how to handle the discontinuities in the symbolic dynamics which occur when the computed partition element boundaries are not disjoint. The method is applied to compute the symbolic dynamics and entropy bounds for the return map of the singular limit of a switching system with hysteresis and the forced Van der Pol equation.

The forced van der Pol equation

We report on a study of the forced van der Pol equation by solving numerically the differential equation for a variety of values of the parameters ϵ,F and ω. In doing so, many striking and interesting trajectories can be discovered and phenomena such as frequency entrainment, almost periodic solutions, space filling trajectories and seemingly chaotic behaviour are explored. These examples naturally give rise to computer laboratory problems suitable for student research and small group projects.

Horseshoes in the forced van der Pol system

The forced van der Pol equation was introduced in the 1920s as a model of an electrical circuit. Cartwright and Littlewood established the existence of invariant sets with complex topology in this system. This paper contains, for the first time, a full description of the nonwandering set for an open set of parameters near the singular limit of the forced van der Pol equation. In particular, we prove the existence of a hyperbolic chaotic invariant set. We also prove that the system is structurally stable. The analysis is conducted from the perspective of geometric singular perturbation theory. Verifying the hypothesis is done by implementing self-validating numerical algorithms.

A method to determine the periodic solution of the non-linear dynamics system

In this paper, the generalized shooting method and the harmonic balancing method to determine the periodic orbit, its period and the approximate analytic expression of the non-linear bearing–rotor system are presented. At first, by changing the time scale, the period of the periodic orbit of the non-linear system is drawn into the governing equation of the system explicitly. Then, the generalized shooting procedure is sought out. The increment value changed in the iteration procedure is selected by using the optimization method. The procedure involves determining the periodic orbit and its period of the system, and the stability of the periodic solution is determined by using Floquet stability theory. The validity of such method is verified by determining the periodic orbit and period of the forced van der Pol equation. Secondly, the periodic solution of the non-linear rotor–bearing system is expanded into Fourier series according to the character of the solution obtained by using the generalized shooting method. Then the approximate analytic expression of the periodic solution of the system is obtained by using the harmonic balancing method. Theoretically, the solution with any precision can be obtained by adding the number of the harmonics. At last, the periodic orbit, period and approximate analytic expressions of the periodic solution of the non-linear rotor–bearing system are provided.

BISTABILITY OF HARMONICALLY FORCED RELAXATION OSCILLATIONS

Relaxation oscillations appear in processes which involve transitions between two states characterized by fast and slow time scales. When a relaxation oscillator is coupled to an external periodic force its entrainment by the force results in a response which can include multiple periodicities and bistability. The prototype of these behaviors is the harmonically driven van der Pol equation which displays regions in the parameter space of the driving force amplitude where stable orbits of periods 2n ± 1 coexist, flanked by regions of periods 2n + 1 and 2n - 1. The parameter regions of such bistable orbits are derived analytically for the closely related harmonically driven Stoker–Haag piecewise discontinuous equation. The results are valid over most of the control parameter space of the system. Also considered are the reasons for the more complicated dynamics featuring regions of high multiple periodicity which appear like noise between ordered periodic regions. Since this system mimics in detail the less analytically tractable forced van der Pol equation, the results suggest extensions to situations where forced relaxation oscillations are a component of the operating mechanisms.

A note on the forced Van der Pol equation

Abstract It is well known that the forced Van der Pol equation e u +(u 2 −1) u +u=α (where u = d u d t and e→0 ) is a singularly perturbed differential equation having exceptional solutions (called canards) for some values of α(e) . It was already proved that there are exactly two distinct complex canards solutions (v+,α+) , (v−,α−) of the transformed equation ev d v d u =(1−u 2 )v+α−u , which are bounded in wide infinite sectors centered at u=0 and containing u=1 . In this Note, explicit asymptotic approximations for α+−α− as e→0 and, as n→∞ , of the an (of the asymptotic series α = ∑ a n e n corresponding to α ) will be proved. The main tool of the proof is the analytic continuation of the solutions v± in a subset of C containing [−1+X l e 1/3 ,+∞[ ⊂ R .

The role of Poincaré—Andronov—Hopf bifurcations in the application of variable-coefficient harmonic balance to periodically forced nonlinear oscillators

The method of variable-coefficient harmonic balance (VCHB) presented in Summers (1995) is applied to the Duffing and the periodically forced van der Pol oscillator equations. A rationale is given of how Poincare-Andronov-Hopf (PAH) bifurcations in the amplitude evolution equations of VCHB relate to the various local bifurcations arising in the solution of these two oscillator equations. In order to demonstrate the crucial role played by PAH bifurcations in the amplitude evolution equations, the theory is applied in its simplest form, that is VCHB with a one-harmonic solution expansion, to the Duffing oscillator equation with a single well and softening-type nonlinearity. A single PAH bifurcation in the amplitude evolution equations is evaluated and the frequency of this bifurcation is then used to construct the curve of symmetry breaking bifurcations of the periodic solutions of the Duffing equation in (ω, F ) control space. In general, the periodic solution is represented by a truncated Fourier series with several harmonics and, therefore, by virtue of the size of the problem, numerical methods are employed to perform the algebra and the analysis. This procedure yields many various eigenvalues to track and, hence, an increased number of PAH bifurcations. The features of nonlinear resonances, period-doubling bifurcations, symmetry breaking bifurcations, subharmonic and superharmonic entrainments and NaimarkSacker bifurcations are tracked throughout control parameter space by tracing the critical eigenvalues associated with the PAH bifurcations. The local bifurcations of the two oscillator equations are classified by evaluating the imaginary part of the critical eigenvalues. In the case of the forced van der Pol equation with order one nonlinearity, the 1:1 and 3:1 superharmonic entrainment boundaries are evaluated, and the symmetry breaking and period-doubling bifurcation boundaries are derived for the first time by a semi-analytic approach.

Numerical Shadowing Near Hyperbolic Trajectories

Shadowing is a means of characterizing global errors in the numerical solution of initial value ordinary differential equations by allowing for a small perturbation in the initial condition. The method presented in this paper allows for a perturbation in the initial condition and a reparameterization of time in order to compute the shadowing distance in the neighborhood of a periodic orbit or more generally in the neighborhood of an attractor. The method is formulated for one-step methods and both a serial and parallel implementation are applied to the forced van der Pol equation, the Lorenz equation and to the approximation of a periodic orbit.

Inverse limits associated with the forced van der Pol equation

In “Qualitative Analysis of the Periodically Forced Relaxation Oscillations,” Mark Levi (Mem. Am. Math. Soc. 32, No. 244, July 1981) gives a nice geometric description of annulus maps associated with the first return map for the forced van der Pol equation ex+Φ(x)x+ex=bp(t). In this paper, it is shown that for certain parameter valuesb, the full attracting sets of the annulus maps described by Levi can be realized as inverse limits of circles. Furthermore, we show that the annulus map is topologically conjugate to the shift homeomorphism on the inverse limit space.

Asymmetric spectral broadening of modulated electrostatic ion‐cyclotron waves

Modulated, current‐driven, electrostatic ion‐cyclotron (CDEIC) waves are shown to exhibit amplitude and frequency modulation, spectral broadening, and time‐averaged frequency pulling. The observed spectral broadening is asymmetric and sensitively dependent on the driving frequency. Qualitative features of the experimental data are reproduced by the forced van der Pol equation and are explainable using processes associated with driven self oscillations. These results may be relevant to ionospheric modification experiments involving the controlled modulation of the natural electrojet.

Analytical conditions for strange chaotic and nonchaotic attractors of the quasiperiodically forced van der Pol equation

We propose an analytical method for predicting the boundaries of the region in parameter space within which a strange nonchaotic attractor exists for a quasiperiodically forced van der Pol equation. One boundary is obtained from the condition for existence of a simple quasiperiodic solution to the equation. The other is shown to be well approximated by the condition for Hopf bifurcation in a suitably averaged system of equation. Good agreement with numerical calculations of Lyapunov exponents is demonstrated.

Analytic predictors for strange non-chaotic attractors

Analytic conditions are used to predict the bounds in parameter space of the regions of existence of a non-chaotic strange attractor for the quasi-periodically forced van der Pol equation. One bound arises from the condition for existence of a simple quasi-periodic response to the forcing; the other appears to be related to the occurence of a Hopf bifurcation in the averaged form of the equation.

Numerical Calculation of Invariant Tori

The problem of computing a smooth invariant manifold for a finite-dimensional dynamical system is considered. In this paper, it is assumed that the manifold can be parameterized over a torus in terms of a subset of the system variables. The approach used here then involves solving a system of partial differential equations subject to periodic boundary conditions. The resulting numerical approach is analyzed and contrasted with some previously tested ones, and several methods of implementation are considered. Numerical results are given for the forced van der Pol equation and for a system of two linearly coupled oscillators.

Subharmonic response and synchronisation for the forced van der Pol equation via Cesari's method

Subharmonic response and synchronisation for the forced van der Pol equation via Cesari's method