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Monotonicity of the period function for some planar differential systems. Part I: Conservative and quadratic systems

100%

Chouikha A.

Applicationes Mathematicae

2005

tom 32

nr 3

305-325

We first examine conditions implying monotonicity of the period function for potential systems with a center at 0 (in the whole period annulus). We also present a short comparative survey of the different criteria. We apply these results to quadratic Loud systems $(L_{D,F})$ for various values of the parameters D and F. In the case of noncritical periods we propose an algorithm to test the monotonicity of the period function for $(L_{D,F})$. Our results may be viewed as a contribution to proving (or disproving) a conjecture of Chicone and Jacobs.

Monotonicity of the period function for some planar differential systems. Part II: Liénard and related systems

100%

Chouikha A.

Applicationes Mathematicae

2005

tom 32

nr 4

405-424

We are interested in conditions under which the two-dimensional autonomous system ẋ = y, ẏ = -g(x) - f(x)y, has a local center with monotonic period function. When f and g are (non-odd) analytic functions, Christopher and Devlin [C-D] gave a simple necessary and sufficient condition for the period to be constant. We propose a simple proof of their result. Moreover, in the case when f and g are of class C³, the Liénard systems can have a monotonic period function in a neighborhood of 0 under certain conditions. Necessary conditions are also given. Furthermore, Raleigh systems having a monotonic (or non-monotonic) period are considered.

Ograniczanie wyników

2 Applicationes Mathematicae

2 Chouikha A.

2 2005

Wyniki wyszukiwania

Monotonicity of the period function for some planar differential systems. Part I: Conservative and quadratic systems

Monotonicity of the period function for some planar differential systems. Part II: Liénard and related systems