Kam theory, Lindstedt series and the stability of the upside-down pendulum
Michele V. Bartuccelli G. Gentile Kyriakos V. Georgiou
Discrete & Continuous Dynamical Systems - A 2003, 9(2): 413-426 doi: 10.3934/dcds.2003.9.413
We consider the planar pendulum with support point oscillating in the vertical direction; the upside-down position of the pendulum corresponds to an equilibrium point for the projection of the motion on the pendulum phase space. By using the Lindstedt series method recently developed in literature starting from the pioneering work by Eliasson, we show that such an equilibrium point is stable for a full measure subset of the stability region of the linearized system inside the two-dimensional space of parameters, by proving the persistence of invariant KAM tori for the two-dimensional Hamiltonian system describing the model.
keywords: Lindstedt series averaging KAM theory stability. vertically driven pendulum perturbation theory nonlinear Mathieu's equation upside-down pendulum
Comparison and convergence to equilibrium in a nonlocal delayed reaction-diffusion model on an infinite domain
Michele V. Bartuccelli S.A. Gourley Y. Kyrychko
Discrete & Continuous Dynamical Systems - B 2005, 5(4): 1015-1026 doi: 10.3934/dcdsb.2005.5.1015
We study a nonlocal time-delayed reaction-diffusion population model on an infinite one-dimensional spatial domain. Depending on the model parameters, a non-trivial uniform equilibrium state may exist. We prove a comparison theorem for our equation for the case when the birth function is monotone, and then we use this to establish nonlinear stability of the non-trivial uniform equilibrium state when it exists. A certain class of non-monotone birth functions relevant to certain species is also considered, namely birth functions that are increasing at low densities but decreasing at high densities. In this case we prove that solutions still converge to the non-trivial equilibrium, provided the birth function is increasing at the equilibrium level.
keywords: Nonlocal infinite domain. comparison reaction diffusion time delay
Length scales and positivity of solutions of a class of reaction-diffusion equations
Michele V. Bartuccelli K. B. Blyuss Y. N. Kyrychko
Communications on Pure & Applied Analysis 2004, 3(1): 25-40 doi: 10.3934/cpaa.2004.3.25
In this paper, the sharpest interpolation inequalities are used to find a set of length scales for the solutions of the following class of dissipative partial differential equations

$u_{t}= -\alpha_{k}(-1)^{k} \nabla^{2k}u+\sum_{j=1}^{k-1}\alpha_{j} (-1)^{j}\nabla^{2j}u+\nabla^{2}(u^{m})+u(1-u^{2p}), $

with periodic boundary conditions on a one-dimensional spatial domain. The equation generalises nonlinear diffusion model for the case when higher-order differential operators are present. Furthermore, we establish the asymptotic positivity as well as the positivity of solutions for all times under certain restrictions on the initial data. The above class of equations reduces for some particular values of the parameters to classical models such as the KPP-Fisher equation which appear in various contexts in population dynamics.

keywords: length scales positivity Dissipative equations interpolation inequalities.

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