Transport in higher dimensional phase spaces
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We use a four dimensional symplectic mapping, the coupled cubic-quadratic map, to provide evidence of Arnol’d Diffusion in phase space. We use the method of frequency analysis for dynamical systems to demonstrate the existence of regular orbits, and show that these orbits enclose weakly chaotic orbits which escape in finite time around the tori. A new collocation method for frequency analysis is employed by adapting it to allow for higher precision results. Arbitrary precision numerics are used to obtain highly accurate orbits for long timescales, and the adapted frequency method is used to obtain highly accurate frequencies of the mapping. We review the method of frequency analysis, demonstrate its effectiveness and accuracy in determining frequencies and finding tori in simple systems and low-dimensional mappings, and extend the results to higher dimensions. In the four dimensional mapping, we find several regular orbits with irrational frequency ratios, indicating the existence of tori in the phase space, as well as interior orbits that escape around these tori.