Analysis of the cubic nonlinearity activator-inhibitor model equation using energy methods

Eric Justh; Advisor: P. S. Krishnaprasad
Intelligent Servosystems Lab

Project Background and Goals

There are a variety of pattern-forming systems, and the properties and generation of patterns in various physical contexts, such as in chemical reactions, have been extensively studied. It might be of interest to tailor an artificial dynamical system for the purpose of creating a control scheme based on pattern-forming-system concepts. The ultimate goal would be to provide a robust way of incorporating both local and global feedback in systems consisting of large numbers of sensors and actuators. Strong mathematical results, even for simple model systems, are essential starting points for the design of such control systems.

By a pattern-forming system, we mean a coupled system of PDEs which passes from a stable equilibrium state to one or more pattern solutions as a bifurcation parameter passes through a critical value. The simplest types of patterns are spatially periodic patterns that are stationary with time, spatially uniform patterns that vary periodically with time, and spatially periodic patterns that vary periodically with time. Pattern-forming systems which have not passed through the bifurcation threshold can produce various dissipative structures, including finite amplitude spike solutions (also called autosolitons), which are stable nonlinear solutions characterized by a significant deviation from the spatially uniform equilibrium state over a spatially localized region. Nonlinearity and energy dissipation are what enable the existence and stability of these localized spike solutions. Certain pattern-forming equations can also possess a bistable regime, in which there are two competing spatially uniform equilibrium states. In such bistable systems, the dynamics of the fronts, or domain boundaries, between the competing spatially uniform equilibrium states is of interest.

The usual approach in trying to analyze pattern-forming systems is to consider simple model equations which give rise to the property under study. In this spirit, one class of equations which has received considerable attention are activator-inhibitor equations, which are a special case of reaction-diffusion equations. The equations considered in this work are a special case of activator-inhibitor equations: the cubic nonlinearity model.


Since dissipation plays a crucial role in the behavior of activator-inhibitor equations, it is natural to try to apply energy methods. There are two distinct types of energy method approaches which can be applied to the cubic nonlinearity model. The first type basically consists of manipulating the dynamical equations so as to obtain L2-norm bounds on solutions and their derivatives. The second type consists of trying to find an energy functional which decreases along trajectories of the dynamics.

Project Results

First, we proved existence and uniqueness of weak solutions, and in the process also showed that solutions depend continuously on the initial data. Next, we proved a dissipativity property: the existence of an absorbing set whose omega-limit-set is in fact a global attractor for the dynamics. Finally, we found that for certain parameter values (related to a ratio of time constants), a radially unbounded Lyapunov functional exists. This Lyapunov functional has the property that its time derivative is less than or equal to zero, and is equal to zero only at equilibrium points of the dynamics. The analysis leading to the existence of the Lyapunov functional is an infinite-dimensional generalization of a corresponding result for systems of ODEs dating back to the 1960s.


Knowing that a Lyapunov functional exists for the cubic nonlinearity model contributes to our understanding of the behavior of this important simple example of a pattern-forming system. That we can explicitly write down the Lyapunov functional is valuable because it means we might be able to see how to couple this system with others while still retaining the desired convergence properties. In fact, what originally motivated this inquiry into the mathematics of the cubic nonlinearity model was to understand what the effects of feedback and control inputs would be in terms of existence and stability properties of solutions.

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Last Updated: March 16, 1998.