On the limit cycle of the van der Pol equation

Odani, Kenzi

Displaying similar documents to “On the limit cycle of the van der Pol equation”

Limit cycles in the equation of whirling pendulum with autonomous perturbation

Hana Lichardová (1999)

Applications of Mathematics

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The two-parameter Hamiltonian system with the autonomous perturbation is considered. Via the Mel’nikov method, existence and uniqueness of a limit cycle of the system in a certain region of a two-dimensional space of parameters is proved.

Heteroclinics for a class of fourth order conservative differential equations

Habets, P., Sanchez, L., Tarallo, M., Terracini, S.

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On the limit cycle of the Liénard equation

Kenzi Odani (2000)

Archivum Mathematicum

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In the paper, we give an existence theorem of periodic solution for Liénard equation $\dot{x} = y - F (x)$ , $\dot{y} = - g (x)$ . As a result, we estimate the amplitude $ρ (μ)$ (maximal $x$ -value) of the limit cycle of the van der Pol equation $\dot{x} = y - μ (x^{3} / 3 - x)$ , $\dot{y} = - x$ from above by $ρ (μ) < 2.3439$ for every $μ \neq 0$ . The result is an improvement of the author’s previous estimation $ρ (μ) < 2.5425$ .

On the number of limit cycles of a generalized Abel equation

Naeem Alkoumi, Pedro J. Torres (2011)

Czechoslovak Mathematical Journal

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New results are proved on the maximum number of isolated $T$ -periodic solutions (limit cycles) of a first order polynomial differential equation with periodic coefficients. The exponents of the polynomial may be negative. The results are compared with the available literature and applied to a class of polynomial systems on the cylinder.