Periodic and subharmonic solutions for duffing equation with a singularity
Zhibo Cheng Jingli Ren
Discrete & Continuous Dynamical Systems - A 2012, 32(5): 1557-1574 doi: 10.3934/dcds.2012.32.1557
This paper is devoted to the existence and multiplicity of periodic and subharmonic solutions for a superlinear Duffing equation with a singularity. In this manner, various preceding theorems are improved and sharpened. Our proof is based on a generalized version of the Poincaré-Birkhoff twist theorem.
keywords: superlinear; singularity harmonic and subharmonic solution Poincaré-Birkhoff twist theorem. Duffing equation
How seasonal forcing influences the complexity of a predator-prey system
Xueping Li Jingli Ren Sue Ann Campbell Gail S. K. Wolkowicz Huaiping Zhu
Discrete & Continuous Dynamical Systems - B 2018, 23(2): 785-807 doi: 10.3934/dcdsb.2018043

Almost all population communities are strongly influenced by their seasonally varying living environments. We investigate the influence of seasons on populations via a periodically forced predator-prey system with a nonmonotonic functional response. We study four seasonality mechanisms via a continuation technique. When the natural death rate is periodically varied, we get six different bifurcation diagrams corresponding to different bifurcation cases of the unforced system. If the carrying capacity is periodic, two different bifurcation diagrams are obtained. Here we cannot get a "universal diagram" like the one in the periodically forced system with monotonic Holling type Ⅱ functional response; that is, the two elementary seasonality mechanisms have different effects on the population. When both the natural death rate and the carrying capacity are forced with two different seasonality mechanisms, the phenomena that arise are to some extent different. The bifurcation results also show that each seasonality mechanism can display complex dynamics such as multiple attractors including stable cycles of different periods, quasi-periodic solutions, chaos, switching between these attractors and catastrophic transitions. In addition, we give some orbits in phase space and corresponding Poincaré sections to illustrate different attractors.

keywords: Predator-prey nonmonotonic functional response seasonal forcing bifurcation diagram complexity
Positive periodic solution for Brillouin electron beam focusing system
Jingli Ren Zhibo Cheng Stefan Siegmund
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 385-392 doi: 10.3934/dcdsb.2011.16.385
An experimental conjecture on the existence of positive periodic solutions for the Brillouin electron beam focusing system $x''+a(1+\cos2t)x=\frac{1}{x}$ for $0 < a < 1$ is proved, using a topological degree theorem by Mawhin.
keywords: Brillouin electron beam focusing system singularity. Liénard equation
Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations
Pedro J. Torres Zhibo Cheng Jingli Ren
Discrete & Continuous Dynamical Systems - A 2013, 33(5): 2155-2168 doi: 10.3934/dcds.2013.33.2155
We analyze the non-degeneracy of the linear $2n$-order differential equation $u^{(2n)}+\sum\limits_{m=1}^{2n-1}a_{m}u^{(m)}=q(t)u$ with potential $q(t)\in L^p(\mathbb{R}/T\mathbb{Z})$, by means of new forms of the optimal Sobolev and Wirtinger inequalities. The results is applied to obtain existence and uniqueness of periodic solution for the prescribed nonlinear problem in the semilinear and superlinear case.
keywords: superlinear uniqueness $2n$-order differential equation. Non-degeneracy semilinear

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