arXiv subjects

Evelyn Sander

Publications and source records attributed to Evelyn Sander.

16 recordsLinked to original sources

arXiv2026

Dynamics of perturbed elliptical billiard tables

Dynamical billiards consist of a particle on a two-dimensional table, bouncing elastically off a boundary curve. The state of the system is given by two numbers: one describing the location along the curve where the bounce occurs, and another describing the incoming angle of the bounce. Successive bounces define a two-dimensional area preserving map, and iterating this map gives a dynamical system first studied by Birkhoff. One of the simplest smooth table shapes is that of an ellipse, in which case the dynamics of the billiard map is completely integrable. The longstanding Birkhoff conjecture is that elliptical tables are the only smooth convex table for which complete integrability occurs. In this spirit, we present an implicit real analytic method for iterating billiard maps on perturbed elliptical tables. This method allows us to compute local stable and unstable manifolds of periodic orbits using the parameterization method. Globalizing these local manifolds numerically provides insight into the dynamics of the table.

math.DS↗

arXiv2025

3D Printing of Invariant Manifolds in Dynamical Systems

Invariant manifolds are one of the key features that organize the dynamics of a differential equation. We introduce a novel approach to visualizing and studying invariant manifolds by using 3D printing technology, combining advanced computational techniques with modern 3D printing processes to transform mathematical abstractions into tangible models. Our work addresses the challenges of translating complex manifolds into printable meshes, showcasing results for the following systems of differential equations: the Lorenz system, the Arneodo-Coullet-Tresser system, and the Langford system. By bridging abstract mathematics and physical reality, this approach promises new tools for research and education in nonlinear dynamics. We conclude with practical guidelines for reproducing and extending our results, emphasizing the potential of 3D-printed manifolds to enhance understanding and exploration in dynamical systems theory.

math.DS↗

arXiv2023

Reliability and robustness of oscillations in some slow-fast chaotic systems

A variety of nonlinear models of biological systems generate complex chaotic behaviors that contrast with biological homeostasis, the observation that many biological systems prove remarkably robust in the face of changing external or internal conditions. Motivated by the subtle dynamics of cell activity in a crustacean central pattern generator (CPG), this paper proposes a refinement of the notion of chaos that reconciles homeostasis and chaos in systems with multiple timescales. We show that systems displaying relaxation cycles while going through chaotic attractors generate chaotic dynamics that are regular at macroscopic timescales and are thus consistent with physiological function. We further show that this relative regularity may break down through global bifurcations of chaotic attractors such as crises, beyond which the system may also generate erratic activity at slow timescales. We analyze these phenomena in detail in the chaotic Rulkov map, a classical neuron model known to exhibit a variety of chaotic spike patterns. This leads us to propose that the passage of slow relaxation cycles through a chaotic attractor crisis is a robust, general mechanism for the transition between such dynamics. We validate this numerically in three other models: a simple model of the crustacean CPG neural network, a discrete cubic map, and a continuous flow.

nlin.CD↗

arXiv2023

Cyclic symmetry induced pitchfork bifurcations in the diblock copolymer model

The Ohta-Kawasaki model for diblock copolymers exhibits a rich equilibrium bifurcation structure. Even on one-dimensional base domains the bifurcation set is characterized by high levels of multi-stability and numerous secondary bifurcation points. Many of these bifurcations are of pitchfork type. In previous work, the authors showed that if pitchfork bifurcations are induced by a simple $\mathbb{Z}_2$ symmetry-breaking, then computer-assisted proof techniques can be used to rigorously validate them using extended systems. However, many diblock copolymer pitchfork bifurcations cannot be treated in this way. In the present paper, we show that in these more involved cases, a cyclic group action is responsible for their existence, based on cyclic groups of even order. We present theoretical results establishing such bifurcation points and show that they can be characterized as nondegenerate solutions of a suitable extended nonlinear system. Using the latter characterization, we also demonstrate that computer-assisted proof techniques can be used to validate such bifurcations. While the methods proposed in this paper are only applied to the diblock copolymer model, we expect that they will also apply to other parabolic partial differential equations.

math.DS↗

arXiv2022

Mathematics of Floating 3D Printed Objects

We explore the stability of floating objects through mathematical modeling and experimentation. Our models are based on standard ideas of center of gravity, center of buoyancy, and Archimedes' Principle. We investigate a variety of floating shapes with two-dimensional cross sections and identify analytically and/or computationally a potential energy landscape that helps identify stable and unstable floating orientations. We compare our analyses and computations to experiments on floating objects designed and created through 3D printing. In addition to our results, we provide code for testing the floating configurations for new shapes, as well as giving details of the methods for 3D printing the objects. The paper includes conjectures and open problems for further study.

physics.class-ph↗

arXiv2022

Equilibrium Validation for Triblock Copolymers via Inverse Norm Bounds for Fourth-Order Elliptic Operators

Block copolymers play an important role in materials sciences and have found widespread use in many applications. From a mathematical perspective, they are governed by a nonlinear fourth-order partial differential equation which is a suitable gradient of the Ohta-Kawasaki energy. While the equilibrium states associated with this equation are of central importance for the description of the dynamics of block copolymers, their mathematical study remains challenging. In the current paper, we develop computer-assisted proof methods which can be used to study equilibrium solutions in block copolymers consisting of more than two monomer chains, with a focus on triblock copolymers. This is achieved by establishing a computer-assisted proof technique for bounding the norm of the inverses of certain fourth-order elliptic operators, in combination with an application of a constructive version of the implicit function theorem. While these results are only applied to the triblock copolymer case, we demonstrate that the obtained norm estimates can also be directly used in other contexts such as the rigorous verification of bifurcation points, or pseudo-arclength continuation in fourth-order parabolic problems.

math.DS↗

arXiv2020

A Computer-Assisted Study of Red Coral Population Dynamics

We consider a 13-dimensional age-structured discrete red coral population model varying with respect to a fitness parameter. Our numerical results give a bifurcation diagram of both equilibria and stable invariant curves of orbits. We observe that not only for low levels of fitness, but also for high levels of fitness, populations are extremely vulnerable, in that they spend long time periods near extinction. We then use computer-assisted proofs techniques to rigorously validate the set of regular and bifurcation fixed points that have been found numerically.

math.DS↗

arXiv2020

Equilibrium Validation in Models for Pattern Formation Based on Sobolev Embeddings

In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.

math.DS↗

arXiv2017

Quantitative Quasiperiodicity

The Birkhoff Ergodic Theorem concludes that time averages, i.e., Birkhoff averages, $Σ_{n=0}^{N-1} f(x_n)/N$ of a function $f$ along a length $N$ ergodic trajectory $(x_n)$ of a function $T$ converge to the space average $\int f dμ$, where $μ$ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We introduce a modified average of $f(x_n)$ by giving very small weights to the "end" terms when $n$ is near $0$ or $N-1$. When $(x_n)$ is a trajectory on a quasiperiodic torus and $f$ and $T$ are $C^\infty$, we show that our weighted Birkhoff averages converge 'super" fast to $\int f dμ$ with respect to the number of iterates $N$, i.e. with error decaying faster than $N^{-m}$ for every integer $m$. Our goal is to show that our weighted Birkhoff average is a powerful computational tool, and this paper illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit accuracy.

math.DS↗

arXiv2017

Solving the Babylonian Problem of quasiperiodic rotation rates

A trajectory $u_n := F^n(u_0), n = 0,1,2, \dots $ is quasiperiodic if the trajectory lies on and is dense in some $d$-dimensional torus, and there is a choice of coordinates on the torus $\mathbb{T}$ for which $F$ has the form $F(θ) = θ+ ρ\bmod1$ for all $θ\in\mathbb{T}$ and for some $ρ\in\mathbb{T}$. There is an ancient literature on computing three rotation rates $ρ$ for the Moon. %There is a literature on determining the coordinates of the vector $ρ$, called the rotation rates of $F$. (For $d>1$ we always interpret $\bmod1$ as being applied to each coordinate.) However, even in the case $d=1$ there has been no general method for computing $ρ$ given only the trajectory $u_n$, though there is a literature dealing with special cases. Here we present our Embedding Continuation Method for computing some components of $ρ$ from a trajectory. It is based on the Takens Embedding Theorem and the Birkhoff Ergodic Theorem. Rotation rates are often called "rotation numbers" and both refer to a rate of rotation of a circle. However, the coordinates of $ρ$ depend on the choice of coordinates of $\mathbb{T}$. We explore the various sets of possible rotation rates that $ρ$ can yield. We illustrate our ideas with examples in dimensions $d=1$ and $2$.

math.DS↗

arXiv2016

Measuring quasiperiodicity

The Birkhoff Ergodic Theorem asserts under mild conditions that Birkhoff averages (i.e. time averages computed along a trajectory) converge to the space average. For sufficiently smooth systems, our small modification of numerical Birkhoff averages significantly speeds the convergence rate for quasiperiodic trajectories -- by a factor of $10^{25}$ for 30-digit precision arithmetic, making it a useful computational tool for autonomous dynamical systems. Many dynamical systems and especially Hamiltonian systems are a complex mix of chaotic and quasiperiodic behaviors, and chaotic trajectories near quasiperiodic points can have long near-quasiperiodic transients. Our method can help determine which initial points are in a quasiperiodic set and which are chaotic. We use our {\bf weighted Birkhoff average} to study quasiperiodic systems, to distinguishing between chaos and quasiperiodicity, and for computing rotation numbers for self-intersecting curves in the plane. Furthermore we introduce the Embedding Continuation Method which is a significantly simpler, general method for computing rotation numbers.

math-ph↗

arXiv2015

Quantitative Quasiperiodicity

The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff averages, $Σ_{n=1}^N f(x_n)/N$ of a function $f$ along an ergodic trajectory $(x_n)$ of a function $T$ converges to the space average $\int f dμ$, where $μ$ is the unique invariant probability measure. Convergence of the time average to the space average is slow. We introduce a modified average of $f(x_n)$ by giving very small weights to the "end" terms when $n$ is near $0$ or $N$. When $(x_n)$ is a trajectory on a quasiperiodic torus and $f$ and $T$ are $C^\infty$, we show that our weighted Birkhoff averages converge "super" fast to $\int f dμ$, {\em i.e.} with error smaller than every polynomial of $1/N$. Our goal is to show that our weighted Birkhoff average is a powerful computational tool, and this paper illustrates its use for several examples where the quasiperiodic set is one or two dimensional. In particular, we compute rotation numbers and conjugacies (i.e. changes of variables) and their Fourier series, often with 30-digit precision.

math.DS↗

arXiv2010

Connecting period-doubling cascades to chaos

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value $μ_2$ of the map at which there is chaos. We show that often virtually all (i.e., all but finitely many) ``regular'' periodic orbits at $μ_2$ are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired -- connected to exactly one other cascade, or solitary -- connected to exactly one regular periodic orbit at $μ_2$. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of $F(μ_2, \cdot)$. Examples discussed include the forced-damped pendulum and the double-well Duffing equation.

nlin.CD↗

arXiv2009

The cascades route to chaos

The presence of a period-doubling cascade in dynamical systems that depend on a parameter is one of the basic routes to chaos. It is rarely mentioned that there are virtually always infinitely many cascades whenever there is one. We report that for one- and two-dimensional phase space, in the transition from no chaos to chaos -- as a parameter is varied -- there must be infinitely many cascades under some mild hypotheses. Our meaning of chaos includes the case of chaotic sets which are not attractors. Numerical studies indicate that this result applies to the forced-damped pendulum and the forced Duffing equations, viewing the solutions once each period of the forcing. We further show that in many cases cascades appear in pairs connected (in joint parameter-state space) by an unstable periodic orbit. Paired cascades can be destroyed or created by perturbations, whereas unpaired cascades are conserved under even significant perturbations.

nlin.CD↗

arXiv2009

Period-doubling cascades for large perturbations of Henon families

The Henon family has been shown to have period-doubling cascades. We show here that the same occurs for a much larger class: Large perturbations do not destroy cascades. Furthermore, we can classify the period of a cascade in terms of the set of orbits it contains, and count the number of cascades of each period. This class of families extends a general theory explaining why cascades occur.

math.DS↗

arXiv2009

Period-doubling cascades galore

The appearance of numerous period-doubling cascades is among the most prominent features of {\bf parametrized maps}, that is, smooth one-parameter families of maps $F:R \times {\mathfrak M} \to {\mathfrak M}$, where ${\mathfrak M}$ is a smooth locally compact manifold without boundary, typically $R^N$. Each cascade has infinitely many period-doubling bifurcations, and it is typical to observe -- such as in all the examples we investigate here -- that whenever there are any cascades, there are infinitely many cascades. We develop a general theory of cascades for generic $F$. We illustrate this theory with several examples. We show that there is a close connection between the transition through infinitely many cascades and the creation of a horseshoe.

math.DS↗