Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation
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| Format: | Preprint |
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2019
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| _version_ | 1866916330515464192 |
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| author | Phatangare, Nanasaheb Masalkar, Krishnat Kendre, Subhash |
| author_facet | Phatangare, Nanasaheb Masalkar, Krishnat Kendre, Subhash |
| contents | In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\dot{X}=\begin{cases} (H_y^+,-H_x^+) & \mbox{if}~y>\varepsilon f(x)\\ (H_y^-,-H_x^-) & \mbox{if}~y<\varepsilon f(x) \end{cases}$ equals to the number of positive zeros of $f$ when at $\varepsilon=0$ the system has a period annulus around the origin. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1912_09795 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation Phatangare, Nanasaheb Masalkar, Krishnat Kendre, Subhash Dynamical Systems In this paper, the general planar piecewise smooth Hamiltonian system with period annulus around the center at the origin is considered. We obtain the expressions for the first order and the second order Melnikov functions of it's general second order perturbation, which can be used to find the number of limit cycles bifurcated from periodic orbits. Further, we have shown that the number of limit cycles of the system $\dot{X}=\begin{cases} (H_y^+,-H_x^+) & \mbox{if}~y>\varepsilon f(x)\\ (H_y^-,-H_x^-) & \mbox{if}~y<\varepsilon f(x) \end{cases}$ equals to the number of positive zeros of $f$ when at $\varepsilon=0$ the system has a period annulus around the origin. |
| title | Bifurcations of limit cycles in piecewise smooth Hamiltonian system with boundary perturbation |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/1912.09795 |