Non-resonance with one-sided superlinear growth for indefinite planar systems via rotation numbers

Chunlian Liu; Chunlian Liu

doi:10.3934/math.2022781

AIMS Mathematics

2022, Volume 7, Issue 8: 14163-14186. doi: 10.3934/math.2022781

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Research article

Non-resonance with one-sided superlinear growth for indefinite planar systems via rotation numbers

Chunlian Liu ^,

School of Science, Nantong University, Nantong 226019, China

Received: 27 February 2022 Revised: 18 May 2022 Accepted: 19 May 2022 Published: 30 May 2022
MSC : 34C25, 34B15, 34D15

We consider the non-resonance with one-sided superlinear growth conditions for the indefinite planar system $ z' = f(t, z) $ from a rotation number viewpoint, and obtain the existence of $ 2\pi $-periodic solutions by applying a rotation number approach together with the Poincaré-Bohl theorem. We allow that the angular velocity of solutions of $ z' = f(t, z) $ is controlled by the angular velocity of solutions of two positively homogeneous and oddly symmetric systems $ z' = L_i(t, z), i = 1, 2 $ on the left half-plane, which have rotation numbers that satisfy $ \rho(L_1) > n/2 $ and $ \rho(L_2) < (n+1)/2 $, and allow $ f(t, z) $ to grow superlinearly on the right half-plane. In order to estimate the rotation angle difference of solutions, we develop a system methodology of "tracking" the angle difference of solutions of the system $ z' = f(t, z) $ on each small interval on the given side under sign-varying conditions.
- periodic solutions,
- non-resonance,
- rotation numbers,
- one-sided superlinear,
- Poincaré-Bohl theorem
Citation: Chunlian Liu. Non-resonance with one-sided superlinear growth for indefinite planar systems via rotation numbers[J]. AIMS Mathematics, 2022, 7(8): 14163-14186. doi: 10.3934/math.2022781

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Abstract

We consider the non-resonance with one-sided superlinear growth conditions for the indefinite planar system $ z' = f(t, z) $ from a rotation number viewpoint, and obtain the existence of $ 2\pi $-periodic solutions by applying a rotation number approach together with the Poincaré-Bohl theorem. We allow that the angular velocity of solutions of $ z' = f(t, z) $ is controlled by the angular velocity of solutions of two positively homogeneous and oddly symmetric systems $ z' = L_i(t, z), i = 1, 2 $ on the left half-plane, which have rotation numbers that satisfy $ \rho(L_1) > n/2 $ and $ \rho(L_2) < (n+1)/2 $, and allow $ f(t, z) $ to grow superlinearly on the right half-plane. In order to estimate the rotation angle difference of solutions, we develop a system methodology of "tracking" the angle difference of solutions of the system $ z' = f(t, z) $ on each small interval on the given side under sign-varying conditions.

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