arXiv subjects

James A. Yorke

Publications and source records attributed to James A. Yorke.

At least 19 recordsLinked to original sources

arXiv2026

Shadow chains and Conley chains for continuous-time semiflows

In a recent series of articles we introduced the concept of "stream of a semiflow. A stream is a closed and transitive binary relation which extends the relation "being on the orbit of" and allows to encode the qualitative behavior of a semiflow into a direct graph. The most important stream of a semiflow is its chain stream, based on Charles Conley's chains. In those previous works we omitted several details and proofs on continuous-time semiflows. In the present work we complement those articles as follows: (i) we provide a full proof of the closedness and transitivity of the chain stream for continuous-time semiflows; (ii) we introduce the concept of ``shadow chain'' for a continuous-time semiflow, based on the Anosov-Sinai-Bowen idea of pseudo-orbit. Shadow chains have the advantage that fit naturally with semiflows arising from differential equations. Our main result is that, although the shadow chain stream and the Conley chain stream are in general distinct as binary relations, they yield the same chain-recurrent set, the same nodes, and the same chain graph whenever the semiflow has strong compact dynamics. While doing this, we also introduce an equivalent definition of recurrent point of a stream in terms of forward-orbit equivalence, which simplifies several arguments below, and we strengthen the definition of s-uniform continuity of a semiflow, fixing a gap in the proof of some important results when the space is not locally compact. This is a radical revision of the first version posted to the arXiv.

math.DS↗

arXiv2026

Many coexisting attractors, a case study of the almost-conservative Hénon map

For dynamical systems in the plane, there can be many periodic attractors coexisting in a bounded region. They become easier to find in systems with small dissipation, which we call ``almost-conservative''. We ask what happens when there are many periodic attractors. That is the vague question we start with. For a test study, we chose the Hénon map with a tiny dissipation. We tuned the other parameter to yield a case with 50 attracting periodic orbits. They have a total of 4259 periodic points. We describe how these orbits can be organized into families. In addition to two low-period orbits, the remaining 48 orbits can be classified into three families, which we describe in detail.

math.DS↗

arXiv2025

What is the graph of a dynamical system?

Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our goal has been to define a meaningful concept of graph of any dynamical system. As a result, we base our definition of ``chain graph'' on ``epsilon-chains'', defining both nodes and edges of the graph in terms of chains. In particular, nodes are often maximal limit sets and there is an edge between two nodes if there is a trajectory whose forward limit set is in one node and its backward limit set is in the other. Our initial goal was to prove that every ``chain graph'' of a dynamical system is, in some sense, connected, and we prove connectedness under mild hypotheses.

math.DS↗

arXiv2025

Streams, Graphs and Global Attractors of Dynamical Systems on Locally Compact Spaces

In a recent article, we introduced the concept of streams and graphs of a semiflow. An important related concept is the one of semiflow with {\em compact dynamics}, which we defined as a semiflow $F$ with a {\em compact global trapping region}. In this follow-up, we restrict to the important case where the phase space $X$ is locally compact and we move the focus on the concept of {\em global attractor}, a maximal compact set that attracts every compact subset of $X$. A semiflow $F$ can have many global trapping regions but, if it has a global attractor, this is unique. We modify here our original definition and we say that $F$ has compact dynamics if it has a global attractor $G$. We show that most of the qualitative properties of $F$ are inherited by the restriction $F_G$ of $F$ to $G$ and that, in case of Conley's chains stream of $F$, the qualitative behavior of $F$ and $F_G$ coincide. Moreover, if $F$ is a continuous-time semiflow, then its graph is identical to the graph of its time-1 map. Our main result is that, for each semiflow $F$ with compact dynamics over a locally compact space, the graphs of the prolongational relation of $F$ and of every stream of $F$ are connected if the global attractor is connected.

math.DS↗

arXiv2025

Streams and Graphs of Dynamical Systems

While studying gradient dynamical systems (DSs), Morse introduced the idea of encoding the qualitative behavior of a DS into a graph. Smale later refined Morse's idea and extended it to Axiom-A diffeomorphisms on manifolds. In Smale's vision, nodes are indecomposable closed invariant subsets of the non-wandering set with a dense orbit and there is an edge from node N to node M if the unstable manifold of N intersects the stable manifold of M. Since then, the decomposition of the non-wandering set was studied in many other settings, while the edges component of Smale's construction has been often overlooked. In the same years, more sophisticated generalizations of the non-wandering set were elaborated first by Auslander in 60s, by Conley in 70s and later by Easton and other authors. In our language, each of these generalizations involves the introduction of a closed and transitive extension of the non-wandering relation, that is closed but not transitive. In the present article, we develop a theory that generalizes at the same time both these lines of research. We study the general properties of closed transitive relations ("streams") containing the space of orbits of a discrete- or continuous-time semi-flow and we argue that these relations play a central role in the qualitative study of DSs. All most studied concepts of recurrence currently in literature can be defined in terms of our streams. Finally, we show how to associate to each stream a graph encoding its qualitative properties. The current revision fixes some proof, adds some missing one and adds some example and clarification.

math.DS↗

arXiv2024

The dynamics of the heterochaos baker maps

The heterochaos baker maps are piecewise affine maps of the unit square or cube introduced in [Nonlinearity 34, 2021, 5744--5761], to provide a hands-on, elementary understanding of complicated phenomena in systems of large degrees of freedom. We review recent progress on a dynamical systems theory of the heterochaos baker maps, and present new results on properties of measures of maximal entropy and the underlying Lebesgue measure. We address several conjectures and questions that may illuminate new aspects of heterochaos and inspire future research.

math.DS↗

arXiv2024

A laminar chaotic saddle within a turbulent attractor

Intermittent switchings between weakly chaotic (laminar) and strongly chaotic (bursty) states are often observed in systems with high-dimensional chaotic attractors, such as fluid turbulence. They differ from the intermittency of a low-dimensional system accompanied by the stability change of a fixed point or a periodic orbit in that the intermittency of a high-dimensional system tends to appear in a wide range of parameters. This paper considers a case where the skeleton of a laminar state $L$ exists as a proper chaotic subset $S$ of a chaotic attractor $X$, that is, $S\ \subsetneq\ X$. We characterize such a laminar state $L$ by a chaotic saddle $S$, which is densely filled with periodic orbits of different numbers of unstable directions. This study demonstrates the presence of chaotic saddles underlying intermittency in fluid turbulence and phase synchronization. Furthermore, we confirm that chaotic saddles persist for a wide range of parameters. Also, a kind of phase synchronization turns out to occur in the turbulent model.

nlin.CD↗

arXiv2023

Extinction of multiple populations and a team of Die-out Lyapunov functions

The extinction of species is a major problem of concern with a large literature. Our investigation gives insight into when species extinctions must occur, with an emphasis on determining which species might possibly die out and on how fast they die out. We investigate a differential equations model for population interactions with the goal of determining when several species (\ie, coordinates of a bounded solution) must die out or ``go extinct'' and must do so exponentially fast. Typically each coordinate represents the population density of a different species. For our main tool, we create what we call ``die-out'' Lyapunov functions. A given system may have several or many such functions, each of which is a function of a different set of coordinates. That die-out function implies that one of the species in its subset must die out exponentially fast -- for almost every choice of coefficients of the system. We create a ``team'' of die-out functions that work together to show that $k$ species must die, where $k$ is determined separately. Secondly, we present a ``trophic'' condition for generalized Lotka-Volterra systems that guarantees that there is a trapping region that is globally attracting. That implies that all solutions are bounded.

math.DS↗

arXiv2023

Structured Systems of Nonlinear Equations

In a "structured system" of equations, each equation depends on a specified subset of the variables. In this article, we explore properties common to "almost every" system with a fixed structure and how the properties can be read from the corresponding connection graph. A solution $p$ of a system $F(p)=c$ is called robust if it persists despite small changes in $F$. We establish methods for determining robustness that depends on the structure, as expressed in the properties of the corresponding directed graph of the structured system. The keys to understanding linear and nonlinear structured systems are subsets of variables that we call forward and backward bottlenecks. In particular, when robustness fails in a structured system, it is due to the existence of a unique "backward bottleneck", that we call a "minimax bottleneck". We present a numerical method for locating the minimax bottleneck. We show how to remove it by adding edges to the graph.

math.CA↗

arXiv2023

Hausdorff dimension of Cantor intersections and robust heterodimensional cycles for heterochaos horseshoe maps

As a model to provide a hands-on, elementary understanding of chaotic dynamics in dimension three, we introduce a $C^2$-open set of diffeomorphisms of $\mathbb R^3$ having two horseshoes with different dimensions of instability. We prove that: the unstable set of one horseshoe and the stable set of the other are of Hausdorff dimension nearly $2$ whose cross sections are Cantor sets; the intersection of the unstable and stable sets contains a fractal set of Hausdorff dimension nearly $1$. As a corollary we detect $C^2$-robust heterodimensional cycles. Our proof employs the theory of normally hyperbolic invariant manifolds and the thicknesses of Cantor sets.

math.DS↗

arXiv2023

The twisted baker map

As a model to provide a hands-on, elementary understanding of "vortex dynamics", we introduce a piecewise linear non-invertible map called a twisted baker map. We show that the set of hyperbolic repelling periodic points with complex conjugate eigenvalues and that without complex conjugate eigenvalues are simultaneously dense in the phase space. We also show that these two sets equidistribute with respect to the normalized Lebesgue measure, in spite of a non-uniformity in their Lyapunov exponents.

math.DS↗

arXiv2022

Piecewise-linear maps with heterogeneous chaos

Chaotic dynamics can be quite heterogeneous in the sense that in some regions the dynamics are unstable in more directions than in other regions. When trajectories wander between these regions, the dynamics is complicated. We say a chaotic invariant set is heterogeneous when arbitrarily close to each point of the set there are different periodic points with different numbers of unstable dimensions. We call such dynamics heterogeneous chaos (or hetero-chaos), While we believe it is common for physical systems to be hetero-chaotic, few explicit examples have been proved to be hetero-chaotic. Here we present two more explicit dynamical systems that are particularly simple and tractable with computer. It will give more intuition as to how complex even simple systems can be. Our maps have one dense set of periodic points whose orbits are 1D unstable and another dense set of periodic points whose orbits are 2D unstable. Moreover, they are ergodic relative to the Lebesgue measure.

math.DS↗

arXiv2022

Robustness of solutions of almost every system of equations

In mathematical modeling, it is common to have an equation $F(p)=c$ where the exact form of $F$ is not known. This article shows that there are large classes of $F$ where almost all $F$ share the same properties. The classes we investigate are vector spaces $\mathcal{F}$ of $C^1$ functions $F:\mathbb{R}^N \to \mathbb{R}^M$ that satisfy the following condition: $\mathcal{F}$ has ``almost constant rank'' (ACR) if there is a constant integer $ρ(\mathcal{F}) \geq 0$ such that rank$(DF(p))=ρ(\mathcal{F})$ for ``almost every'' $F\in \mathcal{F}$ and almost every $p\in\mathbb{R}^N$. If the vector space $\mathcal{F}$ is finite-dimensional, then ``almost every'' is with respect to Lebesgue measure on $\mathcal{F}$, and otherwise, it means almost every in the sense of prevalence, as described herein. Most function spaces commonly used for modeling purposes are ACR. In particular, we show that if all of the functions in $\mathcal{F}$ are linear or polynomial or real analytic, or if $\mathcal{F}$ is the set of all functions in a ``structured system'', then $\mathcal{F}$ is ACR. For each $F$ and $p$, the solution set of $p \in \mathbb{R}^N$ is SolSet$(p):= \{x: F(x)=F(p)\}.$ A solution set of $F(p)=c$ is called robust if it persists despite small changes in $F$ and $c$. The following two global results are proved for almost every $F$ in an ACR vector space $\mathcal{F}$: (1) Either the solution set SolSet$(p)$ is robust for almost every $p\in \mathbb{R}^N$, or none of the solution sets are robust. (2) The solution set SolSet$(p)$ is a $C^\infty$-manifold of dimension $d = N-ρ(\mathcal{F})$. In particular, $d$ is the same for almost every $F \in \mathcal{F}$.

math.CA↗

arXiv2021

The graph of the logistic map is a tower

The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node $A$ to node $B$ if, using arbitrary small perturbations, a trajectory starting from any point of A can be steered to any point of B. In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node A to node B, the unstable manifold of some periodic orbit in A contains points that eventually map onto B. For special parameter values, this tower has infinitely many nodes.

math.DS↗

arXiv2021

Infinite towers in the graph of a dynamical system

Chaotic attractors, chaotic saddles and periodic orbits are examples of chain-recurrent sets. Using arbitrary small controls, a trajectory starting from any point in a chain-recurrent set can be steered to any other in that set. The qualitative behavior of a dynamical system can be encapsulated in a graph. Its nodes are chain-recurrent sets. There is an edge from node A to node B if, using arbitrary small controls, a trajectory starting from any point of A can be steered to any point of B. We discuss physical systems that have infinitely many disjoint coexisting nodes. Such infinite collections can occur for many carefully chosen parameter values. The logistic map is such a system, as we showed in arXiv:2008.08338. To illustrate these very common phenomena, we compare the Lorenz system and the logistic map and we show how extremely similar their bifurcation diagrams are in some parameter ranges.

nlin.CD↗

arXiv2020

Network Deconvolution

Convolution is a central operation in Convolutional Neural Networks (CNNs), which applies a kernel to overlapping regions shifted across the image. However, because of the strong correlations in real-world image data, convolutional kernels are in effect re-learning redundant data. In this work, we show that this redundancy has made neural network training challenging, and propose network deconvolution, a procedure which optimally removes pixel-wise and channel-wise correlations before the data is fed into each layer. Network deconvolution can be efficiently calculated at a fraction of the computational cost of a convolution layer. We also show that the deconvolution filters in the first layer of the network resemble the center-surround structure found in biological neurons in the visual regions of the brain. Filtering with such kernels results in a sparse representation, a desired property that has been missing in the training of neural networks. Learning from the sparse representation promotes faster convergence and superior results without the use of batch normalization. We apply our network deconvolution operation to 10 modern neural network models by replacing batch normalization within each. Extensive experiments show that the network deconvolution operation is able to deliver performance improvement in all cases on the CIFAR-10, CIFAR-100, MNIST, Fashion-MNIST, Cityscapes, and ImageNet datasets.

cs.LG↗

arXiv2019

Population collapse in Elite-dominated societies: A differential equations model without differential equations

The HANDY model of Motesharrei, Rivas, and Kalnay examines interactions with the environment by human populations, both between poor and rich people, i.e., "Commoners" and "Elites". The Elites control the society's wealth and consume it at a higher rate than Commoners, whose work produces the wealth. We say a model is "Elite-dominated" when the Elites' per capita population change rate is always at least as large as the Commoners'. We can show the HANDY model always exhibits population crashes for all choices of parameter values for which it is Elite-dominated. But any such model with explicit equations raises questions of how the resulting behaviors depend on the details of the models. How important are the particular design features codified in the differential equations? In this paper, we first replace the explicit equations of HANDY with differential equations that are only described conceptually or qualitatively - using only conditions that can be verified for explicit systems. Next, we discard the equations entirely, replacing them with qualitative conditions, and we prove these conditions imply population collapse must occur. In particular, one condition is that the model is Elite-dominated. We show that the HANDY model with Elite-dominated parameters satisfies our hypotheses and thus must undergo population collapse. Our approach of introducing qualitative mathematical hypotheses can better show the underlying features of the model that lead to collapse. We also ask how societies can avoid collapse.

math.DS↗

arXiv2019

Unsupervised Learning of Dense Optical Flow, Depth and Egomotion from Sparse Event Data

In this work we present a lightweight, unsupervised learning pipeline for \textit{dense} depth, optical flow and egomotion estimation from sparse event output of the Dynamic Vision Sensor (DVS). To tackle this low level vision task, we use a novel encoder-decoder neural network architecture - ECN. Our work is the first monocular pipeline that generates dense depth and optical flow from sparse event data only. The network works in self-supervised mode and has just 150k parameters. We evaluate our pipeline on the MVSEC self driving dataset and present results for depth, optical flow and and egomotion estimation. Due to the lightweight design, the inference part of the network runs at 250 FPS on a single GPU, making the pipeline ready for realtime robotics applications. Our experiments demonstrate significant improvements upon previous works that used deep learning on event data, as well as the ability of our pipeline to perform well during both day and night.

cs.CV↗