Transient chaos under coordinate transformations in relativistic systems

D. S. Fernández, Á. G. López, J. M. Seoane, and M. A. F. Sanjuán

Phys. Rev. E 101, 062212 – Published 23 June 2020

Abstract

We use the Hénon-Heiles system as a paradigmatic model for chaotic scattering to study the Lorentz factor effects on its transient chaotic dynamics. In particular, we focus on how time dilation occurs within the scattering region by measuring the time with a clock attached to the particle. We observe that the several events of time dilation that the particle undergoes exhibit sensitivity to the initial conditions. However, the structure of the singularities appearing in the escape time function remains invariant under coordinate transformations. This occurs because the singularities are closely related to the chaotic saddle. We then demonstrate using a Cantor-like set approach that the fractal dimension of the escape time function is relativistic invariant. In order to verify this result, we compute by means of the uncertainty dimension algorithm the fractal dimensions of the escape time functions as measured with an inertial frame and a frame comoving with the particle. We conclude that, from a mathematical point of view, chaotic transient phenomena are equally predictable in any reference frame and that transient chaos is coordinate invariant.

Received 9 March 2020
Revised 10 May 2020
Accepted 15 June 2020

DOI:https://doi.org/10.1103/PhysRevE.101.062212

Physics Subject Headings (PhySH)

Chaos Control & applications of chaos

Chaotic systems Dynamical systems

Nonlinear Dynamics

Authors & Affiliations

D. S. Fernández, Á. G. López, J. M. Seoane ^*, and M. A. F. Sanjuán

Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain

^*jesus.seoane@urjc.es

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Issue

Vol. 101, Iss. 6 — June 2020

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Images

Figure 1
(a) Three-dimensional representation of the Hénon-Heiles potential $V (x, y) = \frac{1}{2} (x^{2} + y^{2}) + x^{2} y - \frac{1}{3} y^{3}$ . (b) The isopotential curves in the physical space show that the Hénon-Heiles system is open and has triangular symmetry. If the energy of the particle is higher than a threshold value, related to the potential saddle points, there exist unbounded orbits. Following these trajectories the particle leaves the scattering region through any of the three exits.
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Figure 2
The evolution of a particle launched within the scattering region from the same initial condition for different values of $β$ . (a) For a very low $β$ (Newtonian approximation), the particle is trapped in the KAM tori and describes a bounded trajectory. (b) The value of $β$ is large enough to destroy the KAM tori and the particle leaves the scattering region following a trajectory typical of transient chaos. (c) Finally, a larger value of $β$ than in (b) makes the particle escape more rapidly.
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Figure 3
(a) Each of the exits is identified by a different color: Exit 1 (red), Exit 2 (green), and, finally, Exit 3 (blue). In order to avoid redundant results due to the triangular symmetry of the well, we only let the particle evolve from the angular region $θ_{0} \in [π / 2, 5 π / 6]$ (dashed black lines). (b) Scattering function of the exits $(2000 \times 2000)$ given the parameter map $(β \in [0.5, 0.99], θ_{0} \in [π / 2, 5 π / 6])$ in the hyperbolic regime.
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Figure 4
(a) Lorentz factor evolution $γ (t)$ of three trajectories: a fast escape (yellow curve) and two typical transient chaotic trajectories (red and blue curves). The dashed black guideline represents the Lorentz factor value of $\bar{γ}$ , corresponding to $β = 0.75$ . The time differences $δ t (t)$ along these trajectories is also shown. (b) The scattering function of escape times $t_{e}$ at logarithmic scale given the parameter map $(β \in [0.5, 0.99], θ_{0} \in [π / 2, 5 π / 6])$ . The two dashed black lines correspond to (c) and (d), which show the scattering function of the escape time $t_{e} (θ_{0})$ (blue dots) and $τ_{e} (θ_{0})$ (red dots) for $β = 0.5$ and $β = 0.8$ , respectively. (e, f) Time difference function $δ t_{e} (θ_{0})$ for the same values of $β$ .
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Figure 5
(a) A scheme to visualize the physical effect of a reference frame modification on the unpredictability of the escape times, where $h = 0.005$ . (b) Fractal dimensions according to exits $d_{e}$ (green symbols), escape time $d_{t}$ (blue symbols), and escape proper time $d_{τ}$ (red symbols), with standard deviations computed by the uncertainty dimension algorithm versus 25 equally spaced values of $β \in [0.5, 0.98]$ .
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