Slow divergence integral and its application to classical Liénard equations of degree 5 |
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Authors: | Chengzhi Li Kening Lu |
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Affiliation: | 1. LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China;2. School of Mathematics and Statistics, Xi''an Jiaotong University, Xi''an 710049, China;3. Department of Mathematics, Brigham Young University, Provo, UT 84602, USA |
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Abstract: | The slow divergence integral is a crucial tool to study the cyclicity of a slow–fast cycle for singularly perturbed planar vector fields. In this paper, we deduce a useful form for this integral in order to apply it to various problems. As an example, we use it to prove that the slow divergence integral along any non-degenerate slow–fast cycle for singular perturbations of classical Liénard equations of degree 5 has at most one zero, and the zero is simple if it exists; hence the cyclicity of any non-degenerate slow–fast cycle in this class of equations is at most 2. Up to now there were many interesting results about Liénard equations of degree 3, 4 and ≥6, but almost nothing is known about degree 5. The result in this paper can be seen as a first stage to study the uniform property for classical Liénard equations of degree 5. |
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Keywords: | 34C07 34C08 37G15 |
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