Global Dynamics of a Semilinear Parabolic Equation Driven by a Chaotic System

Abstract

We study a closed interconnection between a semilinear parabolic equation on a bounded domain and a four-dimensional nonlinear oscillator. The finite-dimensional state modulates the parabolic evolution through four time-dependent coefficients, while the parabolic state acts back on the oscillator through a single bounded scalar observable, so that neither component evolves autonomously. Closing the loop changes the analysis in three ways: boundedness of the finite-dimensional component is no longer postulated but derived from a Lyapunov inequality that is required to survive bounded perturbations; the interconnection generates a genuine autonomous semiflow on the product space \(H_0^1(\Omega)\times\mathbb{R}^4\), which removes any recourse to processes or skew-product constructions; and the one-way cascade reappears as the singular limit of vanishing feedback. We prove local and global well-posedness, exhibit a bounded absorbing set, and construct a compact global attractor in the product space, the asymptotic compactness being obtained from a uniform bound in a fractional power space. The same bound yields a uniform precompactness property for the family of attractors, from which we deduce that the attractors converge upper semicontinuously to the attractor of the uncoupled problem as the feedback intensity vanishes.