Hey all. I am wondering if I have a dynamical system that is too expensive to integrate (particularly for certain parameter space), but I do know the equations analytically, is it possible to find its Lyapunov exponents without numerically calculating the phase-space trajectories?
My guess is yes since at each point in the phase space, I can solve the direction and magnitude/speed of the flow, and thus I can advance a small time step. Do this for two close-by points in the phase space for certain time steps, assuming that they are also close to the attractor, I should be able to get the Lyapunov exponents and estimate the dimension of the attractor? My question is I haven't found any method in the literature like I described above, hence I am not sure if I am missing something. Thanks!
Hi Shiqi, I do not have a general answer to your question, but, with several coauthors have published several papers in which Lyapunov exponents were estimated without calculating full trajectories. Usually this involved following a phase space trajectory from some typical initial state (characterized e.g. by a collision between particles) to the next state with similar characteristics. Perhaps you can take some advantage of these.
Lyapunov Exponents and Kolmogorov-Sinai Entropy for the Lorentz Gas at Low Densities [Phys. Rev. Lett. 74, 4412 (1995)]
Lyapunov Exponents from Kinetic Theory for a Dilute, Field-Driven Lorentz Gas
Phys. Rev. Lett., 77:1974, 1996
Lyapunov Spectrum and the Conjugate Pairing Rule for a Thermostatted Random Lorentz Gas, Phys. Rev. Lett. 78, 207 and 211, 1997
The Kolmogorov-Sinai Entropy for Dilute Gases in Equilibrium
Phys. Rev. E 56, 5272 , 1997
Largest Lyapunov Exponent for Many Particle Systems at Low Densities
Phys. Rev. Lett. 80, 2035 (1998)
Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy methods for sums of Lyapunov exponents for dilute gases
Chaos 8, 444 (1998)
Front Propagation Techniques to Calculate the Largest Lyapunov Exponent of Dilute Hard Disk Gases
Journal of Statistical Physics volume 109, pages 641–669 (2002)
Goldstone modes in Lyapunov spectra of hard sphere systems
Phys. Rev. E 70, 016207 (2004)
Lyapunov spectrum of the many-dimensional dilute random Lorentz gas
Phys. Rev. E 70, 036209 (2004)
Thermodynamic formalism for the Lorentz gas with open boundaries in d dimensions
Phys. Rev. E 71, 036213 (2005)
Radius of curvature approach to the Kolmogorov-Sinai entropy of dilute hard particles in equilibrium
J. Stat. Mech., 2011:P08012, 2011
Hi Shiqi,
I think Henk gave you an important clue: "Usually this involved following a phase space trajectory from some typical initial state (characterized e.g. by a collision between particles) to the next state with similar characteristics." As I understand it, Lyapunov exponents characterize a solution (an object in state space) and not the general system per se. Hence you must know *where* to evaluate the local converging and diverging rates in order to, later, average over them. The "where" is usually a trajectory (called the fiducial trajectory). Henk seems to have an alternative solution to this.
Wishing you success!
In fact, our methods rely on having a phase space that is uniform at large scales, plus ergodic, so typical trajectories are dense on this phase space (there may be non-visited islands, which then have to be excluded. These will have their own Lyapunov exponents). Trajectories are supposed to be decomposable into a chain of segments that may be considered as sampled from, say an ensemble of possible segments. The Lyapunov exponents then may be obtained from an average over this ensemble. This is not stated explicitly in these words in the papers, but is in fact what is done.
- Badges
- Science topic
- Similar topics
- Biological Science
- Histology
- Fixatives