Practical Numerical Algorithms for Chaotic Systems by Thomas S. Parker

By Thomas S. Parker

The target of this e-book qre to provide an straightforward creation on chaotic structures for the non-specialist, and to offer and vast package deal of machine algorithms ( within the type of pseudocode) for simulating and characterizing chaotic phenomena. those numerical algorithms have been applied in a software program package deal referred to as INSITE (Interactive Nonlinear process Investigative Toolkit for everybody) that is being allotted individually.

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Thus, the correction term, H(x('»)/(h,/(x(i»)), is the time it would take the trajectory to travel from x(') to ~ if the the vector field were (locally) constant with value /(x('»). Henon's method This clever method is due to Henon [1982]. The technique hinges on his observation that if one wants to know the state of a system at time t, one simply integrates the system to time t. It follows that if the independent variable of the system is X n , the next crossing with the hyperplane Xn = a is calculated by integrating to aj there is no interpolation or time-step halving.

5) where h E IRn is a vector normal to ~, XI: E IRn is some point lying on the hyperplane, and (u, v) := uT v is the inner product. , as in Fig. 3. , (h,f( o. , (h,f( o. ~ in a P±: ~ --+~. --+ ~. --+ ~. P±(x) is the first point where o. P+ and P_ are called one-sided Poincare maps while Pi is called the two-sided Poincare map. , an x E ~ for which (h, f(x)) = 0) satisfies the direction criterion of each of the three maps.

The main drawback of the higher-order Poincare map is that it requires an enormous amount of input data to produce a reasonable amount of output. If one assumes that each sampling reduces the amount of data by a factor of 100, then a third-order Poincare map CHAPTER 2. 0 -5 . 11: Higher-order Poincare map of a three-periodic trajectory with frequency base {I, V2, va}. (a) First-order Poincare map sampled at rate 1; (b) the second-order Poincare map sampled at rates 1 and V2. 4. ALGORITHMS requires 10000 more data points than a first-order map.

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