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Tracking sustained chaos

  • US 6,763,271 B2
  • Filed: 04/12/2001
  • Issued: 07/13/2004
  • Est. Priority Date: 04/12/2001
  • Status: Expired due to Fees
First Claim
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1. A method for controlling the operation of a nonlinear system responsive to parametric signals for sustaining and tracking a chaotic system, said method comprising the steps:

  • (a) representing a nonlinear dynamic system as a function of a system parameter, wherein the function is represented by the formula;

    xn+1=T(xn, δ

    n), and the system parameter is represented by;

    δ

    n

    n

    δ

    n, and an output value;

    (b) generating a parametric signal corresponding to an initial value of the system parameter;

    (c) producing an output signal corresponding to an output value from the nonlinear system in response to the parametric signal;

    (d) calculating iterations of the function; and

    (e) generating a new parametric value if a current iteration falls within a predetermined neighborhood of a previous iteration by calculating x0 and Xn+1 in the formula;

    xn+1

    x0=A(xn

    x0)=B(δ

    n

    δ

    0), where A is the derivative of T with respect to x and B is the derivative of the T with respect to δ

    at (x0, δ

    0); and

    the perturbation of the system parameter is given by the formula;



    n

    δ

    0)=−

    K(x−

    x0) where vector K is such that target point, xn+1 lies inside a neighborhood of an endpoint of a previously existing chaotic transient;

    wherein the new parametric value is applied to the nonlinear dynamic system.

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