Morse decomposition of global attractors with infinite components
Tomás Caraballo Juan C. Jara José A. Langa José Valero
Discrete & Continuous Dynamical Systems - A 2015, 35(7): 2845-2861 doi: 10.3934/dcds.2015.35.2845
In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.
keywords: Morse decomposition infinite components gradient dynamics gradient-like semigroup. Lyapunov function
Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients
Juan C. Jara Felipe Rivero
Discrete & Continuous Dynamical Systems - A 2014, 34(10): 4127-4137 doi: 10.3934/dcds.2014.34.4127
In this work we study the asymptotic behaviour of the following prey-predator system \begin{equation*} \left\{ \begin{split} &A'=\alpha f(t)A-\beta g(t)A^2-\gamma AP\\ &P'=\delta h(t)P-\lambda m(t)P^2+\mu AP, \end{split} \right. \end{equation*} where functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are not necessarily bounded above. We also prove the existence of the pullback attractor and the permanence of solutions for any positive initial data and initial time, making a previous study of a logistic equation with unbounded terms, where one of them can be negative for a bounded interval of time. The analysis of a non-autonomous logistic equation with unbounded coefficients is also needed to ensure the permanence of the model.
keywords: non-autonomous dynamical systems Non-autonomous prey-predator system unbounded time-dependent coefficients evolution processes. pullback attractor permanece of solutions non-autonomous logistic equation

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