Trajectory-free approximation of phase space structures using the trajectory divergence rate

Abstract

This paper introduces the trajectory divergence rate, a scalar field which locally gives the instantaneous attraction or repulsion of adjacent trajectories. This scalar field may be used to find highly attracting or repelling invariant manifolds, such as slow manifolds, to rapidly approximate hyperbolic Lagrangian coherent structures, or to provide the local stability of invariant manifolds. This work presents the derivation of the trajectory divergence rate and the related trajectory divergence ratio for two-dimensional systems, investigates their properties, shows their application to several example systems, and presents their extension to higher dimensions.

Appendix: Derivation of Eq. (17)

Starting with (15), the trajectory-normal repulsion rate, $\rho _T$ can be written, to leading order in T, as,

$$\begin{aligned} \begin{aligned} \rho _T&= 1+ \left( \text {tr}({\mathbf {S}})-\frac{{\mathbf {v}} ^\dagger {\mathbf {S}}{\mathbf {v}}}{\left| {\mathbf {v}}\right| ^2}\right) T\\&= 1+ \frac{1}{\left| {\mathbf {v}}\right| ^2}(\text {tr}({\mathbf {S}}) \left| {\mathbf {v}}\right| ^2-{\mathbf {v}}^ \dagger {\mathbf {S}}{\mathbf {v}})T \\&= 1+ \frac{{\mathbf {v}}^\dagger (\text {tr}({\mathbf {S}}){\mathbf {I}} - {\mathbf {S}}){\mathbf {v}}}{\left| {\mathbf {v}}\right| ^2}T \end{aligned} \end{aligned}$$

(40)

For a 2-tensor, ${\mathbf {A}}$, the relation $\text {tr}({\mathbf {A}}){\mathbf {I}}-{\mathbf {A}}$ in index notation may be written as $A_{ii}\delta _{jk} - A_{jk}$.

$$\begin{aligned} \begin{aligned} \text {tr}({\mathbf {A}}){\mathbf {I}}-{\mathbf {A}}&= A_{ii}\delta _{jk} - A_{jk} \\&= A_{il}\delta _{li}\delta _{jk} - A_{il}\delta _{lk}\delta _{ji} \\&= A_{il}(\delta _{li}\delta _{jk} - \delta _{lk}\delta _{ji}) \\&= A_{il}\varepsilon _{lj}\varepsilon _{ik} \\&= {\mathbf {R}}^\dagger {\mathbf {A}}{\mathbf {R}} \end{aligned} \end{aligned}$$

(41)

where $\varepsilon _{ij}$ is the two-dimensional Levi-Civita symbol which, for a 2x2 matrix, is the index representation of the negative of the $90^\circ$ counter-clockwise rotation matrix, $\varepsilon _{ij} = -{\mathbf {R}}$. Therefore, for small time T, $\rho _T$ may be written as,

$$\begin{aligned} \begin{aligned} \rho _T&= 1+ \frac{{\mathbf {v}}^\dagger ({\mathbf {R}} ^\dagger {\mathbf {S}}{\mathbf {R}}){\mathbf {v}}}{\left| {\mathbf {v}}\right| ^2}T \\&= 1+ \frac{({\mathbf {R}}{\mathbf {v}})^\dagger {\mathbf {S}} ({\mathbf {R}}{\mathbf {v}})}{\left| {\mathbf {v}}\right| ^2}T \end{aligned} \end{aligned}$$

(42)

which can alternatively be written in terms of the unit normal field, ${\mathbf {n}} = {\mathbf {R}}{\mathbf {v}}/\left| {\mathbf {v}}\right|$, as in (3), yielding

$$\begin{aligned} \rho _T = 1+ \langle {\mathbf {n}},{\mathbf {S}}{\mathbf {n}} \rangle T \end{aligned}$$

(43)

which gives the leading order behavior defined by the instantaneous rate,

$$\begin{aligned} {{\dot{\rho }}} = \langle {\mathbf {n}},{\mathbf {S}}{\mathbf {n}} \rangle \end{aligned}$$

(44)

Note that the rate of length change for an infinitesimal material element vector $\ell$ based at ${\mathbf {x}}_0$ and advected under the flow is

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} |\ell | = \frac{1}{| \ell |} \langle \ell ,{\mathbf {S}}\ell \rangle \end{aligned}$$

(45)

Thus, the leading order behavior of the trajectory-normal repulsion rate for short time $T$ can be thought of as the rate of stretching of unit normal vectors, normal to the invariant manifold passing through ${\mathbf {x}}_0$. This value is locally maximized along the most repulsive (or attractive) manifolds, which provide the most influential core of phase space deformation patterns.