2.5 Nonlinear Resonance and Chaos
25
2.4.3 Lattice Surfaces of Section
It is possible to construct surfaces of section for two DoF periodic materials, when
the material is composed of identical unit cells. If one can identify a line of potential
minima (a trench) in the unit cell, then a surface of section can be constructed along
that line of potential energy minima using Birkhoff coordinates (p s , s). However,
in this case the surface of section is a lattice surface of section (LSOS). In a LSOS,
points are plotted each time a trajectory crosses the line of potential energy minima,
regardless of the unit cell traversed by the trajectory. Examples of LSOSs were
constructed in Porter et al. (2017) and Barr et al. (2017) and are discussed in more
detail in Chap. 7.
2.5 Nonlinear Resonance and Chaos
Chaotic regions occur when isolating integrals of motion are destroyed locally by
nonlinear resonances. Walker and Ford (1969) show this explicitly for a simple
model Hamiltonian. Let us first consider the case of a nonlinear system with two
degrees of freedom and with a single resonance between these two degrees of
freedom.
2.5.1 Single-Resonance Hamiltonians
In terms of action-angle variables, a general single-resonance Hamiltonian can be
written
H = H 0 (J 1 , J 2 ) + V n 1 ,n 2 (J 1 , J 2 ) cos(n 1 θ 1 − n 2 θ 2 ) = E,
(2.33)
where (J 1 , J 2 , θ 1 , θ 2 ) are action-angle variables. This system has a second isolating
integral
I = n 2 J 1 + n 1 J 2 = C 2 ,
(2.34)
where C 2 is a constant. It is easy to see that Eq. (2.34) is an isolating integral. We
write Hamilton’s equations of motion for J 1 and J 2 ,
dJ 1
dt
= −
∂H
∂θ 1
= n 1 V n 1 ,n 2 sin(n 1 θ 1 − n 2 θ 2 )
(2.35)
25
2.4.3 Lattice Surfaces of Section
It is possible to construct surfaces of section for two DoF periodic materials, when
the material is composed of identical unit cells. If one can identify a line of potential
minima (a trench) in the unit cell, then a surface of section can be constructed along
that line of potential energy minima using Birkhoff coordinates (p s , s). However,
in this case the surface of section is a lattice surface of section (LSOS). In a LSOS,
points are plotted each time a trajectory crosses the line of potential energy minima,
regardless of the unit cell traversed by the trajectory. Examples of LSOSs were
constructed in Porter et al. (2017) and Barr et al. (2017) and are discussed in more
detail in Chap. 7.
2.5 Nonlinear Resonance and Chaos
Chaotic regions occur when isolating integrals of motion are destroyed locally by
nonlinear resonances. Walker and Ford (1969) show this explicitly for a simple
model Hamiltonian. Let us first consider the case of a nonlinear system with two
degrees of freedom and with a single resonance between these two degrees of
freedom.
2.5.1 Single-Resonance Hamiltonians
In terms of action-angle variables, a general single-resonance Hamiltonian can be
written
H = H 0 (J 1 , J 2 ) + V n 1 ,n 2 (J 1 , J 2 ) cos(n 1 θ 1 − n 2 θ 2 ) = E,
(2.33)
where (J 1 , J 2 , θ 1 , θ 2 ) are action-angle variables. This system has a second isolating
integral
I = n 2 J 1 + n 1 J 2 = C 2 ,
(2.34)
where C 2 is a constant. It is easy to see that Eq. (2.34) is an isolating integral. We
write Hamilton’s equations of motion for J 1 and J 2 ,
dJ 1
dt
= −
∂H
∂θ 1
= n 1 V n 1 ,n 2 sin(n 1 θ 1 − n 2 θ 2 )
(2.35)