# American Institute of Mathematical Sciences

April  2018, 38(4): 2171-2185. doi: 10.3934/dcds.2018089

## Theory of rotated equations and applications to a population model

 1 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2 School of Mathematics Sciences, Qufu Normal University, Qufu 273165, China 3 Department of Mathematics, Shanghai Normal University Shanghai, Shanghai 200234, China 4 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

* Corresponding author: Lijuan Sheng

Received  June 2017 Revised  July 2017 Published  January 2018

Fund Project: The first author is supported by National Natural Science Foundation of China (11431008 and 11771296)

We consider a family of scalar periodic equations with a parameter and establish theory of rotated equations, studying the behavior of periodic solutions with the change of the parameter. It is shown that a stable (completely unstable) periodic solution of a rotated equation varies monotonically with respect to the parameter and a semi-stable periodic solution splits into two periodic solutions or disappears as the parameter changes in one direction or another. As an application of the obtained results, we give a further study of a piecewise smooth population model verifying the existence of saddle-node bifurcation.

Citation: Maoan Han, Xiaoyan Hou, Lijuan Sheng, Chaoyang Wang. Theory of rotated equations and applications to a population model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2171-2185. doi: 10.3934/dcds.2018089
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##### References:
Local behavior of stable or completely unstable periodic solution
Local behavior of semi-stable periodic solution
Asymptotic behavior of solutions for $\lambda>\lambda_0$
Local behavior near a semi-stable periodic solution
Behavior of solutions as $\lambda>\lambda^*$ and $x_0>K$ or $x_0<0$
Behavior of $\varphi_s(t,\lambda)$ and $\varphi_u(t,\lambda)$
Behavior of solutions as $\lambda<0$
Behavior of periodic solutions as $\lambda$ varies
 behavior of periodic solution stable completely unstable upper-stable lower-unstable upper-unstable lower-stable $\frac{\partial f}{\partial \lambda}\geq 0$ increasing with $\lambda$ increasing increasing with $\lambda$ decreasing split with $\lambda$ increasing disappears with $\lambda$ increasing $\frac{\partial f}{\partial \lambda}\leq 0$ increasing with $\lambda$ decreasing increasing with $\lambda$ increasing disappears with $\lambda$ increasing split with $\lambda$ increasing
 behavior of periodic solution stable completely unstable upper-stable lower-unstable upper-unstable lower-stable $\frac{\partial f}{\partial \lambda}\geq 0$ increasing with $\lambda$ increasing increasing with $\lambda$ decreasing split with $\lambda$ increasing disappears with $\lambda$ increasing $\frac{\partial f}{\partial \lambda}\leq 0$ increasing with $\lambda$ decreasing increasing with $\lambda$ increasing disappears with $\lambda$ increasing split with $\lambda$ increasing

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