This paper summarises an investigation of chaos in a toy potential which mimics much of the behaviour observed for the more realistic triaxial generalisations of the Dehnen potentials , which have been used to model cuspy triaxial galaxies both with and without a supermassive black hole .
The potential is the sum of an anisotropic harmonic oscillator potential , V _ { o } = { 1 \over 2 } ( a ^ { 2 } x ^ { 2 } + b ^ { 2 } y ^ { 2 } + c ^ { 2 } z ^ { 2 } ) , and a spherical Plummer potential , V _ { p } = - M _ { BH } / \sqrt { r ^ { 2 } + { \epsilon } ^ { 2 } } , with r ^ { 2 } = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } .
Attention focuses on three issues related to the properties of ensembles of chaotic orbits which impact on chaotic mixing and the possibility of constructing self-consistent equilibria : ( 1 ) What fraction of the orbits are chaotic ?
( 2 ) How sensitive are the chaotic orbits , i.e . , how large are their largest ( short time ) Lyapunov exponents ?
( 3 ) To what extent is the motion of chaotic orbits impeded by Arnold webs , i.e . , how ‘ sticky ’ are the chaotic orbits ?
These questions are explored as functions of the axis ratio a:b:c , black hole mass M _ { BH } , softening length { \epsilon } , and energy E with the aims of understanding how the manifestations of chaos depend on the shape of the system and why the black hole generates chaos .
The simplicity of the model makes it amenable to a perturbative analysis .
That it mimics the behaviour of more complicated potentials suggests that much of this behaviour should be generic .