IJPAM: Volume 92, No. 2 (2014)

NONLINEARLY PERTURBED HEAT EQUATION

Maria de Fátima Correia$^1$, Carlos Ramos$^2$, Sandra Vinagre$^3$
$^{1,2,3}$Department of Mathematics, CIMA-UE
University of Évora
Rua Romão Ramalho 59, 7000-671, Évora, PORTUGAL


Abstract. We consider the linear heat equation with appropriate boundary conditions describing the temperature on a wire with adiabatic endpoints. We also consider a perturbation, which provokes a global change in the temperature of the wire. This perturbation occurs periodically and is modeled by an iterated nonlinear map of the interval belonging to a one-parameter family of quadratic maps, $f_{\mu }$. We observe a long term stabilization, under time evolution, of the number of new critical points of the temperature function. However, for certain values of the parameter $\mu $, even with the stabilization effect of the number of critical points, the evolution of the temperature function is chaotic. We study the parameters of the system, that is, difusion coeficient and $\mu $, in order to characterize the observed behaviour and its dependence on the topological invariants of $f_{\mu }$, in particular the dependence on the chaotic behaviour of $f_{\mu }$.

Received: February 6, 2014

AMS Subject Classification: 37E05, 39B12, 35K05, 37B40, 37C25, 74H65

Key Words and Phrases: heat equation, chaotic dynamics, iteration theory, topological entropy

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DOI: 10.12732/ijpam.v92i2.10 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 92
Issue: 2
Pages: 279 - 296

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).