Global Bifurcations and Fluctuation Spectra of Local Expansion Rates in Nonlinear Dynamical Systems

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Abstract
The definition of the fluctuation spectrum of local expansion rates of nearby orbits is extended to a global region of phase space. It contains information on repellers as well as attractors coexisting in the region. Lyapunov exponents of attractors and repellers, the escape rates for repellers and some structures of these invariant sets are represented by this spectrum. Creations and annihilations of these invariant sets, the change of their nature and interrelations, occurring through global bifurcations when a control parameter value is varied, can be seen clearly in terms of this spectrum. On the basis of the statistical-mechanical formalism for this spectrum some of the invariant sets coexisting in phase space for a fixed value of control parameter are interpreted as in phase equilibrium under the weighted average. At the bifurcation points the present first order phase transitions coincide with those for attractors found in previous papers.

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