Fast Indicators for Orbital Stability: A Survey on Lyapunov and Reversibility Errors

Abstract

We present a survey on the recently introduced fast indicators for Hamiltonian systems, which measure the sensitivity of orbits to small initial displacements, Lyapunov error (LE), and to a small additive noise, reversibility error (RE). The LE and RE are based on variational methods and require the computation of the tangent flow or map. The modified reversibility error method (REM) measures the effect of roundoff and is computed by iterating a symplectic map forward and backward the same number of times. The smoothest indicator is RE since it damps the oscillations of LE. It can be proven that LE and RE grow following a power law for regular orbits and an exponential law for chaotic orbits. There is a numerical evidence that the growth of RE and REM follows the same law. The application to models of celestial and beam dynamics has shown the reliability of these indicators.

Chaos Detection by Fast Dynamic Indicators in Reflecting Billiards

“…In this case, the reversibility error is simply related to the Lyapunov error as Previously, the process has been considered with the noise applied to both the B and F iterations. The covariance matrix of the linear response in in this case is given by and proof of ( 9 ) can be found in [ 3 ] Section 2.3. The asymptotic behavior of the invariants of the reversibility error covariance matrix is determined by the positive Lyapunov exponents.…”

“…Electromagnetic cavities exhibit wave chaos that can be predicted by a semi-classical analysis and random matrix theory; see [ 1 ] for a two-dimensional open electromagnetic cavity, ref. [ 2 ] for closed three-dimensional cavities and [ 3 ] for a review. These powerful prediction tools hold true and can be extended to include coupling through antennas and wave-guides, as well as interconnected cavities [ 4 ].…”

Polynomial stochastic dynamical indicators

“…All three indicators allow one to directly study the effect of uncertainty without the need to run a Monte Carlo simulation and recompute multiple times the value of the chaos indicators. Unlike previous works that aimed at differentiating deterministic chaos from the effect of stochastic processes (Rosso et al 2007;Poon and Barahona 2001;Turchetti and Panichi 2019) or identify particular types of motion from time series (Cincotta et al 1999), in this paper we propose indicators that quantify the effect of parametric uncertainty in the dynamic model. Furthermore, the third indicator, called pseudo-diffusion exponent in the following, is shown to be more computationally advantageous as it does not require the derivation and propagation of the variational equations.…”

Polynomial Stochastic Dynamical Indicators

“…All three indicators allow one to directly study the effect of uncertainty without the need to run a Monte Carlo simulation and recompute multiple times the value of the chaos indicators. Unlike previous works that aimed at differentiating deterministic chaos from the effect of stochastic processes [26][27][28] or identify particular types of motion from time series [29], in this paper we propose indicators that quantify the effect of parametric uncertainty in the dynamic model. Furthermore, the third indicator, called pseudo-diffusion exponent in the following, is shown to be less computationally advantageous as it does not require the derivation and propagation of the variational equations.…”

“…While the accuracy of the computation of the CG tensor degrades with this approach, the authors in [13] points out that: "finite differencing may unveil Lagrange Coherent Structures more reliably than obtaining derivatives of the flow analytically". From (28), one can now introduce the Cauchy-Green Tensor ∆ c ii of the coefficients c i :…”