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Phase-space growth rates, local Lyapunov spectra, and symmetry breaking for time-reversible dissipative oscillators. (English) Zbl 1221.34096

Summary: We investigate and discuss the time-reversible nature of phase-space instabilities for several flows \(\dot x=f(x)\). The flows describe thermostatted oscillator systems in two through eight phase-space dimensions. We determine the local extremal phase-space growth rates, which bound the instantaneous comoving Lyapunov exponents. The extremal rates are point functions which vary continuously in phase space. The extremal rates can best be determined with a ‘singular-value decomposition’ algorithm. In contrast to these precisely time-reversible local ‘point function’ values, a time-reversibility analysis of the comoving Lyapunov spectra is more complex. The latter analysis is nonlocal and requires the additional storing and playback of relatively long (billion-step) trajectories.
All the oscillator models studied here show the same time reversibility symmetry linking their time-reversed and time-averaged ‘global’ Lyapunov spectra. Averaged over a long-time-reversed trajectory, each of the long-time-averaged Lyapunov exponents simply changes signs. The negative or positive sign of the summed-up and long-time-averaged spectra in the forward or backward time direction, respectively, is the microscopic analog of the Second Law of Thermodynamics. This change of the signs of the individual global exponents contrasts with typical more-complex instantaneous ‘local’ behavior, where there is no simple relation between the forward and backward exponents other than the local (instantaneous) dissipative constraint on their sum. As the extremal rates are point functions, they too always satisfy the sum rule.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37D05 Dynamical systems with hyperbolic orbits and sets
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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