Abstract:Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the incr… Show more
“…Having, thus, access to the deviation vectors, we compute the Lyapunov exponents in descending order (i.e., k 1 > k 2 … > k 2N ) following the so-called standard method. [35][36][37] B. Complex statistics shows persisting chaos in the Klein-Gordon chain…”
Section: A the Disordered Quartic Klein-gordon Modelmentioning
We investigate dynamically and statistically diffusive motion in a Klein-Gordon particle chain in the presence of disorder. In particular, we examine a low energy (subdiffusive) and a higher energy (self-trapping) case and verify that subdiffusive spreading is always observed. We then carry out a statistical analysis of the motion in both cases in the sense of the Central Limit Theorem and present evidence of different chaos behaviors, for various groups of particles. Integrating the equations of motion for times as long as 10 9 , our probability distribution functions always tend to Gaussians and show that the dynamics does not relax onto a quasi-periodic KAM torus and that diffusion continues to spread chaotically for arbitrarily long times.
“…Having, thus, access to the deviation vectors, we compute the Lyapunov exponents in descending order (i.e., k 1 > k 2 … > k 2N ) following the so-called standard method. [35][36][37] B. Complex statistics shows persisting chaos in the Klein-Gordon chain…”
Section: A the Disordered Quartic Klein-gordon Modelmentioning
We investigate dynamically and statistically diffusive motion in a Klein-Gordon particle chain in the presence of disorder. In particular, we examine a low energy (subdiffusive) and a higher energy (self-trapping) case and verify that subdiffusive spreading is always observed. We then carry out a statistical analysis of the motion in both cases in the sense of the Central Limit Theorem and present evidence of different chaos behaviors, for various groups of particles. Integrating the equations of motion for times as long as 10 9 , our probability distribution functions always tend to Gaussians and show that the dynamics does not relax onto a quasi-periodic KAM torus and that diffusion continues to spread chaotically for arbitrarily long times.
“…Although a continuous spatially extended system could be discretized in space and thus reduced to the set of ODEs, the result of the direct application of the standard procedures 11,14 to the calculation of the spectrum of Lyapunov exponents depends on the step of discretization. 17 Moreover, the physical meaning of such Lyapunov exponents is vague since their number and the corresponding Lyapunov vectors depend on the method of discretization.…”
Section: General Approach To Calculation Of Lyapunov Exponentsmentioning
confidence: 99%
“…[11][12][13][14] However, a straightforward application of these methods to the spatially extended systems is only possible for the systems which are naturally discrete in space, [15][16][17] e.g., to the arrays of coupled oscillators or maps. Importantly, the direct application of numerical techniques for the calculation of Lyapunov exponents, developed for systems with finite dimension of the phase space, to continuous spatially extended systems is rather ambiguous 17 and unreliable.…”
“…Thus in the spin-orbit problem only three LCEs are independent; it is sufficient to only study those which are positive or zero. (For a review of the mathematical results regarding LCEs see Benettin et al (1980a)). …”
A plot of spin rate versus orientation when Hyperion is at the pericenter of its orbit (surface of section) reveals a large chaotic zone surrounding the synchronous spin-orbit state of Hyperion, if the satellite is assumed to be rotating about a principal axis which is normal to its orbit plane. This means that Hyperion's rotation in this zone exhibits large, essentially random variations on a short time scale. The chaotic zone is so large that it surrounds the 1/2 and 2 states, and libration in the 3/2 state is not possible. Stability analysis shows that for libration in the synchronous and 1/2 states, the orientation of the spin axis normal to the orbit plane is unstable, whereas rotation in the 2 state is attitude stable. Rotation in the chaotic zone is also attitude unstable. A small deviation of the principal axis from the orbit normal leads to motion through all angles in both the chaotic zone and the attitude unstable libration regions. Measures of the exponential rate of separation of nearby trajectories in phase space (Lyapunov characteristic exponents) for these three-dimensional motions indicate the the tumbling is chaotic and not just a regular motion through large angles. As tidal dissipation drives Hyperion's spin toward a nearly synchronous value, Hyperion necessarily enters the large chaotic zone. At this point Hyperion becomes attitude unstable and begins to tumble. Capture from the chaotic state into the synchronous or 1/2 state is impossible since they are also attitude unstable. The 3/2 state does not exist. Capture into the stable 2 state is possible, but improbable. It is expected that Hyperion will be found tumbling chaotically.
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