We summarize hereafter basic results pertaining to Hamiltonian systems that are needed in the exposition of our work. An excellent introduction to the corresponding theory can be found in classical references [14,19].
2.1 Hamiltonian Systems and Area-Preserving Maps
A wide variety of physical systems can be described mathematically through Hamilton’s equations,
dqi
dt D @H
@pi; dpi
dt D @H
@qi: (1)
The state of the system is entirely described by the point z.t/ D .p1; : : : ; pN; q1; : : : ; qN/, in the2N-dimensional phase space. Thepiare the momenta,qi are the positions, and the scalar functionH.p;q; t/is called the Hamiltonian and typically describes the total energy of the system.
These equations lead to an ordinary different equation (ODE):
dz
dt Df.z/; z.0/Dz0; (2)
whereby z0constitute the initial condition. The solution z.t;z0/can be associated with a flow mapftgt2IR, which describes the transport induced in phase space by the dynamical system:
t.z0/WDz.t;z0/; 0.z0/z0: (3) In the following we consider Hamiltonian systems with two degrees of freedom (of the form zD .p1; p2; q1; q2/) that are invariant under the motion. In this case, one of the variables can be expressed as a function of the other three such as p2 D p2.p1; q1; q2; H D E/. This effectively transposes the problem to a three-dimensional coordinate system where the motion is confined to a so-called invariant torus [14]. We then construct the Poincar´e map by first selecting a transverse Poincar´e section˘ D fq2D0gon which points are described by their coordinates x.x; y/ WD .p1; q1/. By following the trajectory from this point, we define the Poincar´e map,P, viaxQ D P.x/, wherex is the first return point of the trajectoryQ emanating from x to the plane˘, see Fig.1.
A fundamental property of Hamiltonian systems compared to other dynamical systems is that the volume of a transported region in the phase space is preserved under the flow map. Hence, the Poincar´e map itself is area-preserving and the vector field describing the transport induced by the map is divergence-free [21]. This observation has profound numerical implications since the interpolation schemes that are typically used in visualization literature will generally not preserve that property of the vector field. We return to this topic in Sect.2.3.
2.2 Qualitative Behavior of Hamiltonian Systems
In the simplest case of Hamiltonian motion (referred to as integrable case), the motion is completely ordered. Specifically, the orbits z.:;z0/are either closed (and therefore correspond to a periodic motion) or they are confined to tori that are themselves invariant under the flow map. In the Poincar´e section these tori form
Saddle (“X-point”)
Fig. 1 Top left: Two iterations of a Poincar´e map. Right: Islands of resonance. Top: Integrable case. Separatrices connect saddle points in Poincar´e map, forming the boundary of an island containing a center point. Bottom: Chaotic case. The connections are replaced by the intersection of stable and unstable manifolds forming the tangles that characterize chaos. Quasi-periodic orbits exist both inside and outside of the island and densely populate KAM manifolds (bottom left)
nested closed curves. At the other extreme of the qualitative spectrum, Hamiltonian systems exhibit ergodic behavior and the motion is random.
The term chaotic systems in contrast refers to systems that are neither fully ordered nor fully chaotic. Unsurprisingly, these systems are typical in practice. A defining objective of their analysis is to understand how the canonical structures of the integrable case progressively break under increasing chaos to give rise to the complex patterns observed in the chaotic regime. Among them, so-called islands appear in the phase portrait and irregular, seemingly random trajectories emerge that wander across circumscribed regions of phase space known as ergodic sea.
Successive iterations of these orbits eventually bring them arbitrarily close to any position within those regions.
2.2.1 Periodic Orbits
By definition, fixed points of the Poincar´e map correspond to periodic orbits.
A periodic orbit of period p returns to its initial position after p iterations of the map: Pp.x0/ WD P.Pp1.x0// D x0. Here p is uniquely defined as the smallest integer that satisfies this relation since the property trivially holds for anyp0 D kp. Similar to what is known from the analysis of critical points in vector fields, the type of a non-degenerate fixed point can be determined by a local linear analysis of the Poincar´e map in its vicinity. This analysis around a
position x0 is based on the properties of the Jacobian Jp WD rxPp, whereby the eigenvaluesi; i D f1; 2gof Jpdetermine the nature of the fixed point. If they are complex conjugates the Poincar´e map is characterized by an elliptical motion near the fixed point. This “O-point” configuration is usually referred to as center in the visualization literature [11]. A local island confining this region of regular motion is present (Fig.1, top right). If the eigenvalues of the Jacobian are real and of opposite sign, x0 is associated with a saddle pattern (or “X-point”) and the eigenvector of Jpassociated with the negative (resp. positive) eigenvalue are tangent to the stable (resp. unstable) manifolds of x0that constitute the boundary of the islands and are the separatrices of the topology. Saddle points and their invariant manifolds are intimately associated with the chaos displayed by Hamiltonian systems. Note that the successive intersections of stable and unstable manifolds form so-called chaotic tangles that shape the dynamics in the chaotic sea, see Fig.1, bottom right.
2.2.2 Quasi-Periodic Orbits and KAM Theory
Beside fixed points, islands, and ergodic seas, the Poincar´e map exhibits curves that are densely covered by quasi-periodic orbits. It follows that the period of these orbits is irrational and the fundamental KAM theorem [1,12,22] states that those KAM surfaces that are “sufficiently irrational” will survive the onset of chaos through nonlinear perturbations. KAM surfaces form perfect transport barriers in the phase portrait, which explains their fundamental importance in physical problems related to confinement. As chaos increases these surfaces are progressively destroyed and replaced by so-called Cantor sets, which offer only partial barrier to transport.
2.3 Numerical Aspects
The analysis of an area preserving map depends heavily on an accurate and efficient integration of the flow mapt t2IR. This computation yields the successive iterates of the Poincar´e mapPi; i 2 f1; ::; Ng. An exception to this rule are discrete analytical maps where an explicit formula f describes the relationship xnC1 D f.xn/. We consider one such map in Sect.5.1. In general, however, the computation of the Poincar´e map is made challenging by the need to maintain long term accuracy in the numerical integration of an ODE. In the context of Hamiltonian systems in particu-lar, the property of area-preservation is essentially impossible to guarantee through conventional integration schemes such as Runge-Kutta methods [18]. So-called geometric (or symplectic) integrators do explicitly enforce the invariance of these properties along the integration [7]. However, their application requires a specific formulation of the dynamics (e.g., an explicit expression for the Hamiltonian of the problem), which is rarely available in numerical simulations. When processing such datasets, the continuous reconstruction of the field through piecewise polynomial functions is not exactly conservative. In this case, the area-preserving property is
a theoretical reference for the behavior of the studied phenomenon rather than a numerical reality. For this work, we applied the divergence cleaning approach based on Hodge projection advocated by Peikert and Sadlo [23]. However we found the overhead caused by the additional piecewise linear divergence-free interpolation they proposed to outweigh the accuracy benefit. Practically, we used in this work the classic Runge-Kutta triple Dormand-Prince DP6(5) method [25] whose dense output provides an excellent balance of accuracy and speed. However, we found it necessary to require very low relative tolerance of the integration scheme ("D108) in order to achieve satisfactory results.