## A perturbation-incremental method for the calculation of semi-stable limit cycles of strongly nonlinear oscillators.(English)Zbl 0964.65145

The authors describe the existing qualitative results regarding the limit cycles for the nonlinear oscillator equations $$x''+g(x)=\lambda f(x,x',\mu)x'$$. To apply a numerical approach, the authors transform this equation into an integral equation, then the initial solution for $$\lambda\sim 0$$ is obtained by using the perturbation method. The solution for an arbitrary value of $$\lambda$$ can be determined by using the incremental approach. Two examples (including the generalized Lienard equation) are demonstrated.

### MSC:

 65P40 Numerical nonlinear stabilities in dynamical systems 65L05 Numerical methods for initial value problems involving ordinary differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 37M05 Simulation of dynamical systems 37C75 Stability theory for smooth dynamical systems 34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
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### References:

 [1] Leung, Journal of Shock and Vibration 1 pp 233– (1994) [2] Leung, Shock and Vibration 2 pp 307– (1995) [3] Leung, Journal of Sound and Vibration 213 pp 907– (1998) · Zbl 1235.34116 [4] Leung, Journal of Sound and Vibration 217 pp 795– (1998) · Zbl 1235.34125 [5] Method of Normal Form Wiley: New York, 1993. [6] Theory of Limit Cycles. Transactions of the Mathematical Monographs, vol. 66. American Mathematical Society: Providence, RI, 1992. [7] Qualitative Theory of Differential Equations. Transactions of the Mathematical Monographs, vol. 101. American Mathematical Society: Providence, RI, 1992. [8] Chan, International Journal of Non-linear Mechanics 31 pp 59– (1996) · Zbl 0864.70015 [9] Xu, Nonlinear Dynamics 11 pp 213– (1996) [10] Xu, Journal of Sound and Vibration 174 pp 563– (1994) · Zbl 0945.70534 [11] Rychkov, Differentsialnye Uravneniya 11 pp 390– (1975) [12] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer: New York, 1983. · Zbl 0515.34001
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