The aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle xo of least period T-0 > 0 when it is perturbed by a small parameter, T-1-periodic perturbation. In the case when T-0/T-1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation epsilon > 0 sufficiently small, the existence of klT(0)-periodic solutions x(epsilon) of the perturbed system which converge to the trajectory (x) over tilde (0) of the limit cycle as epsilon -> 0. Moreover, we state conditions under which T = klT(0) is the least period of the periodic solutions x(E). We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T-0/T-1 is an irrational number we show the nonexistence, whenever T > 0 and epsilon > 0, of T-periodic solutions x(E) of the perturbed system converging to (x) over tilde (0). The employed methods are based on the topological degree.
M., K., O., M., Nistri, P. (2006). Periodic solutions of periodically perturbed planar autonomous systems: A topological approach. ADVANCES IN DIFFERENTIAL EQUATIONS, 11(4), 399-418.
Periodic solutions of periodically perturbed planar autonomous systems: A topological approach
NISTRI, PAOLO
2006-01-01
Abstract
The aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle xo of least period T-0 > 0 when it is perturbed by a small parameter, T-1-periodic perturbation. In the case when T-0/T-1 is a rational number l/k, with l, k prime numbers, we provide conditions to guarantee, for the parameter perturbation epsilon > 0 sufficiently small, the existence of klT(0)-periodic solutions x(epsilon) of the perturbed system which converge to the trajectory (x) over tilde (0) of the limit cycle as epsilon -> 0. Moreover, we state conditions under which T = klT(0) is the least period of the periodic solutions x(E). We also suggest a simple criterion which ensures that these conditions are verified. Finally, in the case when T-0/T-1 is an irrational number we show the nonexistence, whenever T > 0 and epsilon > 0, of T-periodic solutions x(E) of the perturbed system converging to (x) over tilde (0). The employed methods are based on the topological degree.File | Dimensione | Formato | |
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https://hdl.handle.net/11365/17228
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