The Forced van der Pol Equation I: The Slow Flow and Its Bifurcations

The forced van der Pol oscillator has been the focus of scientific scrutiny for almost a century, yet its global bifurcation structure is still poorly understood. In this paper, we present a hybrid system consisting of the dynamics of the trajectories on the slow manifold coupled with "jumps" at the folds in the critical manifold to approximate the fast subsystem. The global bifurcations of the fixed points and periodic points of this hybrid system lead to an understanding of the bifurcations in the periodic orbits (without canards) of the forced van der Pol system.

[1]  M. L. Cartwright.  I. Van der Pol’s Equation for Relaxation Oscillations , 1953 .

[2]  Idan Segev,et al.  Methods in Neuronal Modeling , 1988 .

[3]  J. E. Littlewood,et al.  On Non‐Linear Differential Equations of the Second Order: I. the Equation y¨ − k(1‐y2)y˙ + y = bλk cos(λl + α), k Large , 1945 .

[4]  F. Takens.  Constrained equations; a study of implicit differential equations and their discontinuous solutions , 1976 .

[5]  M. L. Cartwbight.  Balthazar Van Der Pol , 1960 .

[6]  J. Grasman.  Asymptotic Methods for Relaxation Oscillations and Applications , 1987 .

[7]  A. Selverston,et al.  Modeling the gastric mill central pattern generator of the lobster with a relaxation-oscillator network. , 1993, Journal of neurophysiology.

[8]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[9]  John Guckenheimer,et al.  Numerical Computation of Canards , 2000, Int. J. Bifurc. Chaos.

[10]  F. Stumpers,et al.  Balth. van der Pol's Work on Nonlinear Circuits , 1960 .

[11]  J. E. Littlewood,et al.  On Non-Linear Differential Equations of the Second Order: II. The Equation .. y + kf(y, . y + g(y, k) = p(t) = p 1 (t) + kp 2 (t); k > 0, f(y) > 1 , 1947 .

[12]  J. Flaherty,et al.  Frequency Entrainment of a Forced van der pol Oscillator. , 1977 .

[13]  M. Levi.  Qualitative Analysis of the Periodically Forced Relaxation Oscillations , 1981 .

[14]  Ulrich Parlitz,et al.  BIFURCATION STRUCTURE OF THE DRIVEN VAN DER POL OSCILLATOR , 1993 .

[15]  P. Szmolyan,et al.  Canards in R3 , 2001 .

[16]  S. Smale.  Diffeomorphisms with Many Periodic Points , 1965 .

[17]  B. V. D. Pol,et al.  Frequency Demultiplication , 1927, Nature.

[18]  B. V. D. Pol.  The Nonlinear Theory of Electric Oscillations , 1934 .

[19]  J. Keener.  Chaotic behavior in piecewise continuous difference equations , 1980 .

[20]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[21]  A. Y. Kolesov,et al.  Asymptotic Methods in Singularly Perturbed Systems , 1994 .

[22]  J. Tattersall,et al.  The Mathematical Collaboration of M. L. Cartwright and J. E. Littlewood , 1996 .

[23]  N. Levinson,et al.  A Second Order Differential Equation with Singular Solutions , 1949 .

[24]  É. Benoît,et al.  Canards et enlacements , 1990 .

[25]  John Guckenheimer,et al.  The Forced van der Pol Equation II: Canards in the Reduced System , 2003, SIAM J. Appl. Dyn. Syst..

[26]  Julyan H. E. Cartwright,et al.  Dynamics of elastic excitable media , 1999, chao-dyn/9905035.

[27]  M. L. Cartwright.  On non-linear differential equations of the second order , 1949 .

[28]  F. Takens.  Forced oscillations and bifurcations , 2001 .

[29]  V. Arnold.  Dynamical systems V. Bifurcation theory and catastrophe theory , 1994 .

[30]  N. K. Rozov,et al.  Differential Equations with Small Parameters and Relaxation Oscillations , 1980 .

[31]  J. E. Littlewood,et al.  Addendum to 'On Non-Linear Differential Equations of the Second Order, II' , 1949 .

[32]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[33]  J. E. Littlewood.  On non-linear differential equations of the second order: IV. The general equation $$\ddot y + kf\left( y \right)\dot y + g\left( y \right) = bkp\left( \varphi \right)$$ , φ=t+α, φ=t+α , 1957 .