Structural Stability, Catastrophe Theory, and Applied Mathematics The John von Neumann Lecture, 1976 by René Thom Presented by Edgar Lobaton

René Thom (Sep 2, 1923 – Oct 25, 2002) Born in Montbéliard, France His early work was on differential geometry He then worked on singularity theory Developed Catastrophe Theory between 1968-1972 Received the Fields Medal in 1958

Catastrophe Theory (CT) CT emphasizes the qualitative aspect of empirical situations The truth is that CT is not a mathematical theory, but a ‘body of ideas’, I daresay a ‘state of mind’.

Dynamical Systems Dynamics Equilibrium Points For discrete systems these are also called fixed points. For example, Newton Iteration for finding roots of polynomials. We can extend this definition to Attractor sets.

Finding Basins of Attractions For example we can analyze the Julia set for which we have a discrete system with a rational function on the right hand side. We Iterate and color the initial condition based on the attractor set to which it converges The RHS has parameters to set, we consider fixed values for these computations.

Bifurcations There are parameters to fix in our dynamics. What happens when we change their values? An example is the one-hump function

Looking for Equilibrium We are interested on solving the following equation as a function of the parameters In control we do this by use of the Implicit Function Theorem, with the following assumption on the Jacobian

Catastrophe Theory Control Theory then tries to define functions u(t) in such a way as to have the corresponding functions x(t) satisfy some optimality condition. Elementary Catastrophe Theory deals with the cases when the Jacobian of F is singular. From a control point of view, we are interested on staying on the regions where we the Jacobian is non-singular. The plot is an example of “finite codimension type”

Structural Stability The singularity shown before is “unavoidable”, i.e., structurally stable under any small Ck deformation of the equation F(x,u) = 0. However, for something like F(x,u) = u-exp(-1/x2) where we have a flat contact line, the singularity is avoidable (structurally unstable) by a small deformation of F.

Structural Stability (cont…) Another example is Linear Systems If the linear system has no purely imaginary eigenvalues then the system is structurally stable On the other hand, linear systems with purely imaginary eigenvalues are structurally unstable.

Structural Stability (cont…) It is a known result that for a system with finite dimensional states and parameters, there exists only a finite number of “unavoidable” singularities. The general idea in Catastrophe Theory is to study these “unavoidable” catastrophe situations.

The Potential Our assumption is that for any attractor of any dynamical system there is a local Lyapunov function. Hence, the equilibrium point is a solution of an optimality principle.

Final Remarks We still know very little about the global problem of catastrophe theory, which is the problem of dynamic synthesis, i.e. how to relate into a single system a field of local dynamics From my viewpoint, C.T. is fundamentally qualitative, and has as its fundamental aim the explanation of an empirical morphology.

REFERENCES R. Thom, Structural Stability, Catastrophe Theory, and Applied Mathematics. 1976 http://en.wikipedia.org/wiki/Catastrophe_theory http://en.wikipedia.org/wiki/Julia_set http://perso.orange.fr/l.d.v.dujardin/ct/eng_index.html M. Demazure, Bifurcations and Catastrophes, 2000. (pp 1-11) S. Sastry, Nonlinear Systems: Analysis, Stability and Control, 1999. (pp 4-10)