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Pré-Publication, Document De Travail Année : 2007

Inducing periodicity and chaos by negative feedback

Résumé

Suppose that a parameter-dependent system has a stable equilibrium state for each parameter value in an interval and suppose that the equilibrium state has a first order dependence on the parameter. Basing ourselves on a mathematical result published earlier, we propose an algorithm to construct a negative feedback on the parameter which induces a Shilnikov or Lorenz type of chaos by building of homoclinic orbits. The algorithm is robust against variations of parameters and numerical uncertainty. One outcome is, for instance, in the Lorenz type of situation, trajectories visiting, in a chaotic order, neighborhoods of four given stable states of the initial system. We illustrate the results on the three-variables Goodwin model with negative regulation. The second class of results presented here apply to a wider class of systems. Only supposing that a parameter-dependent system has stable states for two values of the parameter, we give a very simple feedback formula which allows us to obtain trajectories oscillating indefinitely between the stable states. The method is simulated on the three-variables Goodwin model with positive or negative regulation and low cooperativity index ($<8$). The simulations show periodic orbits, which bifurcate by a cascade of period doubling when the distance between the stable states is varied, leading to chaos.

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Dates et versions

hal-00194648 , version 1 (07-12-2007)

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Elisabeth Pecou, Bruno Cessac, Guillermo Espinoza. Inducing periodicity and chaos by negative feedback. 2007. ⟨hal-00194648⟩
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