Chaos in the near-horizon dynamics of the dyonic $\rm{AdS_4}$-Reissner-Nordström black hole

TL;DR

This work analyzes chaotic motion of a massless probe particle confined near the horizon of a dyonic $ m{AdS_4}$-RN black hole, treating the total energy $E$, chemical potential $\mu$, and magnetic field $B$ as independent controls. By deriving the near-horizon Hamiltonian with harmonic confinement and employing Poincaré sections and maximum Lyapunov exponents, it reveals a counteracting regulation: at low energy $\Gamma$-driven nonlinearity enhances chaos and can violate the bound $\lambda_L\le\kappa$, while at high energy chaos is suppressed along the extremal line $\Gamma=3$, producing a corridor of regular dynamics. The extremal limit induces qualitative changes in the near-horizon dynamics, notably turning the horizon-induced exponential instability into a softer, power-law behavior. These results connect black hole thermodynamics to microscopic chaotic dynamics, offering new insights for AdS/QCD and nonlinear dynamics in strongly curved spacetimes.

Abstract

We investigate the chaos in the dynamics of a probe massless particle confined by the harmonic potential near the horizon of the dyonic $\rm{AdS_4}$-Reissner-Nordström black hole. The total energy of the particle, chemical potential and magnetic field in this system serving as independently adjustable parameters tune nonlinearity and phase-space structure. By analyzing the trajectories on the Poincaré section and evaluating the Lyapunov exponents, we obtain the dynamical phase diagrams of the chaos and find their counteracting regulatory role: at low energy, chaos is enhanced and the Lyapunov exponent $λ_L$ violates its upper bound (i.e. surface gravity) in the extremal black hole limit(combined paramete $Γ=3$); at high energy, the same extremal limit suppresses chaos, with $λ_L$ dropping to zero and a regular dynamical corridor emerging along $Γ=3$ in the dynamical phase diagrams. These results establish a direct mapping between black hole thermodynamics and microscopic chaos, offering new insights into the AdS/QCD correspondence and nonlinear dynamics in strongly curved spacetimes.

Figures (6)

• Figure 1: Evolution of the blackening factor $f(z)$ with respect to the parameter $\Gamma = \mu^2 z_h^2 + B^2 z_h^4$. The surface shows the value of $f(z)$, while the red dashed trajectory marks the position of the inflection point $(f"(z) = 0)$. When $\Gamma < 1$, the inflection point lies outside the physical region $(0 < z < z_h)$; when $\Gamma > 1$, it enters the physical region, indicating a pronounced distortion in the functional shape. The blue plane marks the critical value $\Gamma = 1$, at which the inflection point is located precisely at the horizon $z = z_h$.
• Figure 2: The Poincaré sections of the particle motion near the black hole horizon for different energies with $\mu=B=0$. As the energy increases, the Kolmogorov-Arnold-Moser (KAM) tori tend to break.
• Figure 3: The Poincaré sections of the particle motion near the black hole horizon for different magnetic fields with $\mu=0, E=8$. As the magnetic field increases, the Kolmogorov-Arnold-Moser (KAM) tori tend to break.
• Figure 4: The Poincaré sections of the particle motion near the black hole horizon for different chemical potentials with $B=0, E=7$. As the chemical potential increases, the Kolmogorov-Arnold-Moser (KAM) tori tend to break.
• Figure 5: Numerical analysis of the maximum Lyapunov exponents (MLEs). After a period of time, the MLEs converge to a set of values.