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Exact Multiplicity and Stability of Periodic Solutions for Duffing Equation with Bifurcation Method

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Abstract

Under some \(L^p\)-norms(\(p\in [1,\infty ]\)) assumptions for the derivative of the restoring force, the exact multiplicity and the stability of \(2\pi \)-periodic solutions for Duffing equation are considered. The nontrivial \(2\pi \)-periodic solutions of it are positive or negative, and the bifurcation curve of it is a unique reversed S-shaped curve. The class of the restoring force is extended, comparing with the class of \(L^{\infty }\)-norm condition. The proof is based on the global bifurcation theorem, topological degree and the estimates for periodic eigenvalues of Hill’s equation by \(L^p\)-norms(\(p\in [1,\infty ]\)).

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Acknowledgements

The author would like to express sincere thank to the anonymous referee whose valuable comments greatly improve the manuscript.

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  1. School of Mathematics, Jilin University, Changchun, 130012, People’s Republic of China

    Shuqing Liang

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Correspondence to Shuqing Liang.

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Supported by National Natural Science Foundation of China Grant No. 11501240.

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Liang, S. Exact Multiplicity and Stability of Periodic Solutions for Duffing Equation with Bifurcation Method. Qual. Theory Dyn. Syst. 18, 477–493 (2019). https://doi.org/10.1007/s12346-018-0296-x

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