DCDS
Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions
Alberto Boscaggin Fabio Zanolin
Discrete & Continuous Dynamical Systems - A 2013, 33(1): 89-110 doi: 10.3934/dcds.2013.33.89
We study the problem of existence and multiplicity of subharmonic solutions for a second order nonlinear ODE in presence of lower and upper solutions. We show how such additional information can be used to obtain more precise multiplicity results. Applications are given to pendulum type equations and to Ambrosetti-Prodi results for parameter dependent equations.
keywords: parameter dependent equations. lower and upper solutions subharmonic solutions Periodic solutions Poincaré-Birkhoff twist theorem
DCDS
Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity
Chiara Zanini Fabio Zanolin
Discrete & Continuous Dynamical Systems - A 2012, 32(11): 4045-4067 doi: 10.3934/dcds.2012.32.4045
We prove the existence of multiple periodic solutions as well as the presence of complex profiles (for a certain range of the parameters) for the steady-state solutions of a class of reaction-diffusion equations with a FitzHugh-Nagumo cubic type nonlinearity. An application is given to a second order ODE related to a myelinated nerve axon model.
keywords: Grindrod and Sleeman model. subharmonics chaotic-like dynamics Nagumo type equations Reaction-diffusion equations periodic solutions stationary solutions
DCDS
Bounded solutions of nonlinear second order ordinary differential equations
Alessandro Fonda Fabio Zanolin
Discrete & Continuous Dynamical Systems - A 1998, 4(1): 91-98 doi: 10.3934/dcds.1998.4.91
We prove the existence of bounded solutions to second order differential equations of Liénard type under asymptotic conditions generalizing recent results of Ahmad and Ortega.
keywords: Second order ordinary differential equations bounded solutions.
DCDS
A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations
Tongren Ding Hai Huang Fabio Zanolin
Discrete & Continuous Dynamical Systems - A 1995, 1(1): 103-117 doi: 10.3934/dcds.1995.1.103
The existence of periodic solutions for some planar systems is investigated. Applications are given to positive solutions for a class of Kolmogorov systems generalizing a predator - prey model for the dynamics of two species in a periodic environment.
keywords: Kolmogorov systems Lotka-Volterra equations positive periodic solutions continuation theorem.

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