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2026-06-11 23:58

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Conference: Bucharest University Faculty of Physics 2026 Meeting

Section: Theoretical and Computational Physics, Applied Mathematics

Title:

Critical phenomena and universality at the transition to chaos in conservative dynamical systems

Authors:

Gabriel MAJERI (1), Virgil BĂRAN (1)

^*
Affiliation:

1) University of Bucharest

E-mail

gabriel.majeri@unibuc.ro, virgil.baran@unibuc.ro

Keywords:

dynamical systems, chaos theory, nonlinear dynamics, Lyapunov exponents, universal behavior, critical phenomena

Abstract:

Physical systems described by nonlinear equations often exhibit chaotic trajectories. Chaos limits our ability to understand and predict a system’s long-term evolution. Several methods were developed for (numerically) studying these phenomena over longer periods of time: Lyapunov exponents, Poincaré surface-of-section, Generalized Alignment Index (GALI), Smaller Alignment Index (SALI) etc. These tools reveal the complex process through which chaos occurs in various systems and can be also used for conservative (Hamiltonian) dynamical systems. However, they primarily use “local” information, under the assumption that almost all trajectories visit the entire phase space as the system becomes ergodic. In the presence of regions with invariant tori which stay intact, as predicted by the Kolmogorov-Arnold-Moser (KAM) theorem, these measurements may no longer reflect the general behaviour of the system. This motivated us to initiate a quantitative and global study of chaos in a conservative dynamical system (the Hénon–Heiles model together with a fourth order anharmonic perturbation term), by numerically evaluating the Lyapunov exponents for many initial conditions, for different values of the control system parameters. We identify a power-law behaviour at the transition to chaos, near the critical values of the parameters, from which we try to deduce a universality principle, in analogy to the one observed in dissipative systems.

References:

Strogatz, Steven H. 2024. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. 3rd ed. Chapman and Hall/CRC. https://doi.org/10.1201/9780429398490.

Benettin, Giancarlo, Luigi Galgani, and Jean-Marie Strelcyn. 1976. “Kolmogorov Entropy and Numerical Experiments.” Physical Review A 14 (6): 2338–45. https://doi.org/10.1103/PhysRevA.14.2338.

A.N. Kolmogorov. 1954. “On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian”, Dokl. akad. nauk SSSR, (98), 527–530.

Henon, Michel, and Carl Heiles. 1964. “The Applicability of the Third Integral of Motion: Some Numerical Experiments.” The Astronomical Journal 69 (February): 73. https://doi.org/10.1086/109234.

Baran, V., and A. A. Raduta. 1998. “Classical and Quantal Chaos Described by a Fourth Order Quadrupole Boson Hamiltonian.” International Journal of Modern Physics E 07 (04): 527–51. https://doi.org/10.1142/S0218301398000282.

“An Algorithm for Least-Squares Estimation of Nonlinear Parameters | SIAM Journal on Applied Mathematics.” Journal of the Society for Industrial and Applied Mathematics. https://epubs.siam.org/doi/10.1137/0111030.

Feigenbaum, Mitchell J. 1978. “Quantitative Universality for a Class of Nonlinear Transformations.” Journal of Statistical Physics 19 (1): 25–52. https://doi.org/10.1007/BF01020332.

Acknowledgement:

The authors thank the University of Bucharest for providing them with access to the high-performance computing (HPC) cluster of the University’s Advanced Computing Center (ACC-UB).