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.
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.