Small Periodic Perturbations of an Autonomous System with a Stable Orbit

1998 ◽  
pp. 141-151
Author(s):  
Norman Levinson
Keyword(s):  
Autonomous System ◽  
Stable Orbit ◽  
Periodic Perturbations ◽  
Small Periodic

scholarly journals Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium

2019 ◽  
Vol 18 (4) ◽  
pp. 1695-1709
Author(s):  
Pablo Amster ◽  
◽  
Mariel Paula Kuna ◽  
Gonzalo Robledo ◽  
Keyword(s):  
Multiple Solutions ◽  
Autonomous System ◽  
Periodic Perturbations

scholarly journals Small periodic perturbations of autonomous self-oscillating planar systems

2004 ◽  
Vol 37 (12) ◽  
pp. 363-366
Author(s):  
Mikhail Kamenskii ◽  
Oleg Makarenkov ◽  
Paolo Nistri
Keyword(s):  
Periodic Perturbations ◽  
Planar Systems ◽  
Small Periodic

scholarly journals On resonances under quasi-periodic perturbations of systems with a double limit cycle, close to two-dimensional nonlinear Hamiltonian systems

2021 ◽  
Vol 23 (1) ◽  
pp. 11-27
Author(s):  
Olga S. Kostromina
Keyword(s):  
Numerical Calculation ◽  
Limit Cycle ◽  
Autonomous System ◽  
Theoretical Study ◽  
Type Equation ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Resonant Structures ◽  
Periodic Perturbations ◽  
Double Limit

The effect of multi-frequency quasi-periodic perturbations on systems close to twodimensional nonlinear Hamiltonian ones is studied. It is assumed that the corresponding perturbed autonomous system has a double limit cycle. Analysis of the Poincar´e–Pontryagin function constructed for the autonomous system makes it possible to establish the presence of such a cycle. When the condition of commensurability of the natural frequency of the corresponding unperturbed Hamiltonian system with the frequencies of the quasi-periodic perturbation is fulfilled, the unperturbed level becomes resonant. Resonant structures essentially depend on whether the selected resonance levels coincide with the levels that generate limit cycles in the autonomous system. An averaged system is obtained that describes the topology of the neighborhoods of resonance levels. Possible phase portraits of the averaged system are established near the bifurcation case, when the resonance level coincides with the level in whose neighborhood the corresponding autonomous system has a double limit cycle. To illustrate the results obtained, the results of a theoretical study and of a numerical calculation are presented for a specific pendulum-type equation under two-frequency quasi-periodic perturbations.

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