Abstract
Abstract Following the latest work of Wu et al., we construct time-transformed explicit symplectic schemes for a Hamiltonian system on the description of charged particles moving around a deformed Schwarzschild black hole with an external magnetic field. Numerical tests show that such schemes have good performance in stabilizing energy errors without secular drift. Meantime, tangent vectors are solved from the variational equations of the system with the aid of an explicit symplectic integrator. The obtained tangent vectors are used to calculate several chaos indicators, including Lyapunov characteristic exponents, fast Lyapunov indicators, a smaller alignment index, and a generalized alignment index. It is found that the smaller alignment index and generalized alignment index are the fastest indicators for distinguishing between regular and chaotic cases. The smaller alignment index is applied to explore the effects of the parameters on the dynamical transition from order to chaos. When the positive deformation factor and angular momentum decrease, or when the energy, positive magnetic parameter, and the magnitude of the negative deformation parameter increase, chaos easily occurs for the appropriate choices of initial conditions and the other parameters.
Highlights
Abundant highly nonlinear general relativistic dynamical models greatly enrich the content of chaotic dynamics
The chaotic behavior of Newtonian dynamic systems can be identified by using different methods, such as the Poincaré section method, the Lyapunov index, the local Lyapunov index and its spectral distribution (Contopoulos et al 1999; Szezech et al 2005), the fast Lyapunov index (Froeschle et al 1997; Froeschle & Lega 2000), the spectrum analysis method (Laskar 1994), a relatively limited-time Lyapunov index (Sandor et al 2004), a smaller alignment index (Skokos et al 2004), a power spectrum (Binney & Spergel 1982), and a fractal graph (Levin 2000)
It is called generalized alignment index (GALI), which is a generalization of smaller alignment index (SALI)
Summary
Abundant highly nonlinear general relativistic dynamical models greatly enrich the content of chaotic dynamics. The chaotic behavior of Newtonian dynamic systems can be identified by using different methods, such as the Poincaré section method, the Lyapunov index, the local Lyapunov index and its spectral distribution (Contopoulos et al 1999; Szezech et al 2005), the fast Lyapunov index (Froeschle et al 1997; Froeschle & Lega 2000), the spectrum analysis method (Laskar 1994), a relatively limited-time Lyapunov index (Sandor et al 2004), a smaller alignment index (Skokos et al 2004), a power spectrum (Binney & Spergel 1982), and a fractal graph (Levin 2000) Each of these methods has its advantages and disadvantages, and should be selected carefully.
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