Structural Stability, Catastrophe Theory, and Applied Mathematics

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Structural Stability, Catastrophe Theory, and
Applied Mathematics

• The John von Neumann Lecture, 1976 by René Thom
• Presented by Edgar Lobaton

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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

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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.

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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.

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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.

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Bifurcations

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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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)