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Stable manifolds and predictability of dynamical systems. (English) Zbl 0982.37012
The authors aim at creating a global picture of predictability of a dynamical system, where predictability is measured in terms of effective Lyapunov exponents. These concepts are explained in detail and it is argued that level contours of predictability are close to the stable manifolds of the most unstable set of nonwandering points. By a simple model system the authors illustrate these ideas. A detailed numerical study of a driven damped pendulum and the Lorenz model shows the utility of the predictability concept. In the former, predictability is modulated by unstable periodic orbits, in the latter by the stable manifold of a fixed point. By means of predictability, the dynamically dominant sets can be visualized. The predictability of chaotic transients for parameters of the Lorenz model where no chaotic attractor exists can be captured and predictability for values beyond the homoclinic bifurcation is given in terms of Bessel functions.

MSC:
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37G25 Bifurcations connected with nontransversal intersection in dynamical systems
37D10 Invariant manifold theory for dynamical systems
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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