Discrete-time dynamics, step-skew products, and pipe-flows

TL;DR

The work develops a universal framework for approximating deterministic ergodic dynamics by stochastic-like and continuous-time systems. By building a step-skew product (finite-state Markov driving a deterministic disk dynamics) and a continuous-time perturbed pipe-flow, the authors prove a weak conditional convergence in law between the discrete-time, stochastic description and the continuous-time, deterministic realization. They introduce junctions and pipes to realize a network flow, and demonstrate that the exit statistics and path-space distributions of the pipe-flow can be made arbitrarily close to those of the step-skew model, via zero-noise limits and strong mixing principles. The results imply fundamental indistinguishability between certain classes of deterministic and stochastic time-series and provide a constructive method to simulate and compare such dynamics in a unified, topological framework.

Abstract

A discrete-time deterministic dynamical system is governed at every step by a predetermined law. However the dynamics can lead to many complexities in the phase space and in the domain of observables that makes it comparable to a stochastic process. This behavior is characterized by properties such as mixing and ergodicity. This article presents two different approximation schemes for such a dynamical system. Each scheme approximates the ergodicity of the deterministic dynamics using different principles. The first is a step-skew product system, in which a finite state Markov process drives a dynamics on Euclidean space. The second is a continuous-time skew-product system, in which a deterministic, mixing flow intermittently drives a deterministic flow through a topological space created by gluing cylinders. This system is called a perturbed pipe-flow. We show how these three representations are interchangeable. It is proved that the distribution induced on the space of paths by these three types of dynamics can be made arbitrarily close to each other. This indicates that it is impossible to decide whether a general timeseries is generated by a deterministic or stochastic process, and is of continuous or discrete time.

Figures (7)

• Figure 1: Alternative descriptions of ergodic dynamics. The starting point of this analysis is an ergodic system, as defined in Assumption \ref{['A:f']}. This is a general means of describing most deterministic physical phenomenon. One of its key components is the invariant measure $\mu$. It has two significances -- its support represents the phase space of the dynamics under observation. Its distribution determines the statistical properties of its generated data, as well as its dynamic complexity. The article presents how the dynamics can be approximated by the other two types of dynamics, labelled II and III. Type II is a skew product system in which an autonomous finite-state Markov process drives a dynamical system on $d$-dimensional disks. Type II performs a dual stochastic and topological approximation of Type I. Type III is a deterministic flow through topological spaces called cells and cylinders. A trajectory is observed when it exits these cells. The timeseries created by the series of exit points provide a statistical approximation of the Type II. The meaning of these approximations is explained in the smaller white boxes. They connect various secondary characteristic of these dynamical systems.
• Figure 2: A partial semi-flow (left) and an axial semi-flow (right). These serve as templates for the type of constructions described in the paper. A partial semi-flow generalizes the notion of a semi-flow by allowing trajectories to be finite-time instead of extending indefinitely into the future. Each point $x$ on the phase space $\mathcal{X}$ has an exit time $\mathcal{T}$. The points on $\mathcal{X}$ for which $\mathcal{T} \equiv 0$ are called the exit points. The flow $\Phi$ maps every point $a$ along with its exit time $\mathcal{T} (a)$ into an exit point. Given any time $s$ less than $\mathcal{T} (a)$, $\Phi$ maps $(a,s)$ into a point at time $s$ further along the orbit of $a$. The space $\mathcal{X}$ can be any topological space, and has been drawn as a line in the left figure for simplicity. When $\mathcal{X}$ is indeed an interval and the exit time varies linearly, one gets an interval semi-flow \ref{['eqn:def:intrvl_part_flow']}. An axial semi-flow is a semi-flow which can be projected into the interval semi-flow. Such a flow has been depicted on the right hand diagram.
• Figure 3: Construction of a junction. A $k$-junction in $d$-dimensions is a gluing of $k+1$$d$-dimensional disks, as shown in the figure. See Section \ref{['sec:junction']} for a description of its use. Any such junction represents a state $s$ of the skew-product system \ref{['eqn:Mrkv:1']}, which has $k$ possible outgoing states. The left most face is interpreted as the entry point, and the other $k$ terminals are interpreted as exit windows. See Figure \ref{['fig:frontal']} (a) for the construction for general $k$. This entire topological space can be embedded in $\mathbb{R}^{d+3}$, as explained in the text.
• Figure 4: An frontal view along the axis of a junction. Figure \ref{['fig:cell']} presented a lateral view of a $k$ junction. To place the construction in $\mathbb{R}^{d+3}$ one starts by constructing the following branched 1-manifold in $\mathbb{R}^3$. The first branch representing the input channel, is a straight line segment from $(-1,0,0)$ to $(0,0,0)$. For each $1\leq i \leq k$, the $i$-th output channel is the straight line from $(0,0,0)$ to $(1, \cos \left( \frac{i-1}{k} 2\pi \right), \sin \left( \frac{i-1}{k} 2\pi \right))$. Part (a) presents a frontal view of this branched 1-manifold. One next takes a Cartesian product with $D^d$ as shown in Figure \ref{['fig:cell']} to construct the junction. Every $k$-junction in $d$-dimensions constructed in Figure \ref{['fig:cell']} is provided a vector field. The vector field can be decomposed into two components - axial and lateral. The figure on the right describes the axial component of the flow. The diagram assumes $k=2$ for the sake of simplicity. The axial vector field is kept constant and equal to $1$ throughout the length of the junction. See Figure \ref{['fig:cell_lateral']} for a description of the lateral component of the vector field.
• Figure 5: Lateral flow along a junction. The $k$-junction in $d$-dimensions constructed in Figure \ref{['fig:cell']} is provided a vector field. The vector field can be decomposed into two components - axial and lateral. The figure presents a top-view of the junction, for the simple case when $k=2$. The axial coordinate is represented by a variable $l \in [0,2]$. the branching occurs at $l=1$. The lateral vector field is zero for $l\in [0,1.1]$. In the interval $[1.1,2]$ it is setup so that each of the branches are attracting sets, while the central axis remains neutral. Within the window $[1, 1.1]$ the junction receives a drift from an external mixing flow, shown in red. The input is weighted by the function $w$ from \ref{['eqn:def:weight']}. Note that any trajectory under the combined action of the axial and lateral vector fields travels along the central axis till the branching point $l=1$. Within the window $1\leq l \leq 1.1$ it deviates to either of the branches. Due to the mixing nature of $(\Omega, \Gamma^t)$ and the uncertainty in its initial condition, this is a random event. Beyond the point $l=1.1$ the trajectory gets pulled to that branch in whose basin it lies.

Theorems & Definitions (10)

• Corollary 1
• Corollary 2
• Corollary 3
• Proposition 4: Step-skew approximation of dynamics
• Lemma 3.1
• Lemma 3.2
• Lemma 4.1
• Lemma 5.1
• Lemma 6.1
• Theorem 5