I am trying to understand the behaviour of an Hamiltonian system I'm simulating. I will give a quick context setting. The system is defined as $$ \mathcal{H}(\mathbf{z};\mathbf{z}^*) = \sum_{i=1}^{M}c_i\exp\left(-\frac{(\mathbf{z}^*-\boldsymbol{\zeta}_i^*)^T(\mathbf{z}-\boldsymbol{\zeta})}{2 \sigma^2}\right) $$ where $\mathbf{z} \in \mathbb{C}^n$ is a vector of complex numbers. $\boldsymbol{\zeta}_i \in \mathbb{C}^n \text{ } \forall i$, $c_i \in \mathbb{R} \text{ } \forall i$ and $\sigma \in \mathbb{R}$ are parameters which define the system. The dimension I'm studying is $n=2$, i.e. $\mathbf{z} \in \mathbb{C}^2$. For the sake of numerical computation, I had to convert the real-valued complex Hamiltonian to a decomposed real Hamiltonian as $$\mathbf{z} = (z_1, z_2) \rightarrow \mathbf{X} = (x_1, x_2, x_3, x_4),$$ where $z_1 = x_1 + ix_2$ and $z_2 = x_3 + ix_4$. The Hamiltonian structure remains the same, as given by: $$ \mathcal{H}(\mathbf{X}) = \sum_{i=1}^{M}c_i\exp\left(-\frac{(\mathbf{X}-\tilde{\boldsymbol{\zeta }}_i)^T(\mathbf{X}-\tilde{\boldsymbol{\zeta}})}{2 \sigma^2}\right) $$ I treat this as an Hamiltonian system by considering $(x_1, x_2)$ and $(x_3, x_4)$ as canonical conjugate pairs. The equation of motion can therefore be written as $$\frac{d\mathbf{X}}{dt} = -2 \mathbf{J} \nabla \mathcal{H},$$ where $\mathbf{J}$ is the Poisson matrix, which for the above definition of conjugate pairs becomes $$ \mathbf{J} = \begin{pmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ \end{pmatrix} $$ For the case of $\mathbf{z} \in \mathbb{C}^1$, which is a 1-degree of freedom system, the system does not show chaos because the number of invariants (which is 1, the Hamiltonian) is equal to the DoF (Degree of Freedom) - from the Liouville sense of integrability. For my case of n=2 degrees of freedom, the system should be able to exhibit chaos if there is no additional invariants of motion.
The problem
I was able to find instances of chaos for certain configuration of the Hamiltonian. For the case when the location of the 'wells' defined by the $\boldsymbol{\zeta}_i$'s occur along a 'line' in any of the 6 linear axes, there was no instance of chaos and only periodic behaviour - I verified this by calculating maximum Lyapunov exponents over a subset of the phase space around the location of the 'wells', and they were of the order 1e-6 for integration time of order 1e5. The structure of the Hamiltonian in this case is shown in the figure below.
I reasoned the lack of chaos as due to integrability of the system, because there is a hyper-cylindrical symmetry in the structure of the Hamiltonian, and that must contribute an invariant of the system making the total number of invariants equal to the degrees of freedom. To verify/counter-verify this I tried a system configuration without such symmetry as shown in the following figure:
In such a configuration, I was able to observe chaos with positive Lyapunov exponents of order 1e-2 for up to integration time scales of the order 1e6. My choice of introducing asymmetry was random at first, but later I checked with asymmetry along other 5 planes too, and found out that asymmetry only in the mixed pair variables produced chaos, i.e for the second and third columns of the plots, whereas, asymmetry in configuration for the original Argand plane $(x_1, y_1)$ and $(x_2, y_2)$ (first column of the plots) did not produce chaos and had periodic behaviour. This puts me at my wits end while trying to explain the instances of chaos for these specific configurations. Initially I was under the (blind) impression that asymmetry in any plane should exhibit chaos. But it turns out asymmetry only in non-conjugate pair planes exhibit chaos. How is this explained by the Hamiltonian dynamics of the system? More specifically:
How do I explain the relation between the asymmetry in the non-conjugate pair axes and chaos?
If it is indeed the kind of 'symmetry' that is considered in the Liouville integrability notion, how to obtain the corresponding action-angle variables coordinates? Before I discovered the lack of chaos in the first column of planes, I figured that the Hamiltonian can be written in a hyper-cylindrical coordinate system with 1 angle and 3 distances to explicitly show the symmetry, but even in that case not sure how to arrive at the invariants of motion. But anyway that notion seems to be invalid now because of the periodic behaviour when asymmetry occurs in the first column planes.
Apologies if the post is wordy, but it was inevitable for I feared it lacking crucial details otherwise.
P.S: Plots use older notation but the translation is $(x_1, y_1, x_2, y_2) \rightarrow (x_1, x_2, x_3, x_4)$
