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The Periodic Solution of Van Der Pol's Equation
SIAM Journal on Applied Mathematics, 1998Summary: The author gives the periodic solution of the van der Pol equation \(\ddot{x} + x = \varepsilon(1 - x^2)\dot{x}\) in the form of a series converging for all values of the damping parameter \(\varepsilon\). For \(\varepsilon\) small the series solution reduces to a perturbation series in powers of \(\varepsilon\) and is obtained from this ...
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Dynamics of the Van der Pol equation
IEEE Transactions on Circuits and Systems, 1980This paper is a review of approaches to understanding "chaotic" dynamics in the forced Van der Pol equation. In addition, it discusses the phenomena of entrainment and phase locking from the point of view of dynamical systems theory. There are two principal regions of the parameter space where chaotic motion has been analyzed.
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Theoretical and Mathematical Physics, 2021
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Periodic solutions for van der Pol equation with time delay
Applied Mathematics and Computation, 2007zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Guojian Lin
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Ultradiscrete analogue of the van der Pol equation
Nonlinearity, 2022Abstract Ultradiscretization with parity variables (pUD) is a method of constructing a conditional piecewise linear equation which approximates a given difference equation. Its application is not limited to subtraction-free equation, being different from the traditional ultradiscretization.
Shin Isojima, Seiichiro Suzuki
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Applied Scientific Research, 1968
The family of trajectories (in the phase plane) of van der Pol's equation is a one parameter family of curves; no explicit analytical expression for it is known. The same is true for the orthogonal trajectories of this family. The differential equation of the orthogonal curves is solved here exactly in terms of modified Bessel functions.
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The family of trajectories (in the phase plane) of van der Pol's equation is a one parameter family of curves; no explicit analytical expression for it is known. The same is true for the orthogonal trajectories of this family. The differential equation of the orthogonal curves is solved here exactly in terms of modified Bessel functions.
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Complex dynamics in Duffing–Van der Pol equation
Chaos, Solitons & Fractals, 2006zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Jing, Zhujun, Yang, Zhiyan, Jiang, Tao
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Stable Subharmonics of the Forced Van der Pol Equation
IMA Journal of Applied Mathematics, 1983On etudie numeriquement l'equation de Van der Pol a terme de forcage periodique pour un domaine de valeurs des parametres.
El-Abbasy, E. M., James, Eleanor M.
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A novel approach for solving the Van Der Pol equation
2019 12th International Symposium on Computational Intelligence and Design (ISCID), 2019We transform the second ordinary differential Van Der Pol equation into the first order differential equation system. By this system, we construct an unconstraint optimization problem. Moreover, we give a Rounge-Kutta based Bat algorithm to solve this optimization problem.
Lipu Zhang, Xiaoping Zhang, Yongping Sun
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The Attractor of Perturbed van der Pol's Equation
1983Results of the author concerning attractors of systems close to autonomous ones [Acta Sci. Math. 44, 329-334 (1982; Zbl 0509.34052)] are applied to the second order equation \(u''+m(1-u^ 2)u'+u=F(t,u,u')\), \(m>0\) where F is considered as a bounded perturbation.
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