arXiv subjects

Eran Igra

Publications and source records attributed to Eran Igra.

10 recordsLinked to original sources

arXiv2026

When Entropy flows: drifting along the route to Chaos

Consider a smooth one-parameter family of vector fields defined over some smooth manifold transitions from order into chaos. Inspired by the Second law of Thermodynamics, one is led to ask: can we find a flow whose dynamics realize this transition? To answer this question, motivated by the Mallet-Yorke Orbit Index theory, the Arnold-Khesin scheme for hydrodynamics and a heuristic argument by Rene Thom, we introduce a construction that transforms any one-parameter family of vector fields into a new object: the "Entropy flow". The Entropy flow is a flow defined on the product of the phase space with the parameter space and is best thought of as a flow generated by the original one-parameter family together with a drift in the parameter space, that pushes the trajectory of a given initial condition into a disordered, more complex state. To exemplify, for the Period Doubling, the Ruelle-Takens-Newhouse and the Intermittency routes to chaos the Entropy flow behaves exactly as expected - that is, it truly pushes trajectories into more complex states. In addition, in the spirit of Forcing Theory, in the paper we use the Conley index to discuss how one can use the Entropy flow to study the connection between topology and bifurcations. Moreover, drawing on the numerical and analytic evidence, we will analyze how the Entropy flow behaves in several examples of famous flows, including the Lorenz system, the R\"ossler attractor, and the breakup of the Shilnikov homoclinic scenario.

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arXiv2025

Kneading the Lorenz attractor

A Lorenz map $f:[0,1]\to[0,1]$ is a piecewise continuous map, modeled after an idealized version of the Lorenz attractor. In this paper we settle the following question - how much of the dynamics of the Lorenz attractor can be modeled by such one-dimensional model? In this paper we will prove there exist open regions in the parameter space of the Lorenz system where one can canonically reduce the dynamics of the Lorenz attractor into those of a symmetric Lorenz map $F_\beta$ with a constant slope $\beta\in(1,2]$. As we will show, not only the map $F_\beta$ encodes many of the essential features of the Lorenz attractor, it also governs many of its bifurcations. As such, our results correlate closely with the results of numerical studies, and possibly explain the bifurcation phenomena observed in the Lorenz attractor.

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arXiv2025

Using Erd\H{o}s's methods to study Yorke's problems

In this paper, we study the possible bifurcations of periodic orbits by analyzing them as graphs. In detail, we construct a collection of graphs which are an idealized versions of bifurcation diagrams and color them using the Mallet-Yorke Orbit Index (and the Lefschetz Fixed Point Theorem). The aforementioned allows to study the genericity of routes to chaos, as well as to gain insight into their possible complexity. In particular, our results can be interpreted as saying that there is no upper bound on the possible complexity of routes to chaos in high dimensional systems.

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arXiv2025

Period-Doubling Cascades Invariants: Braided Routes To Chaos

By a classical result of Kathleen Alligood and James Yorke we know that as we isotopically deform a map $f:ABCD\to\mathbb{R}^2$ to a Smale horseshoe map we should often expect the dynamical complexity to increase via a period--doubling route to chaos. Inspired by this fact and by how braids force the existence of complex dynamics, in this paper we introduce three topological invariants that describe the topology of period--doubling routes to chaos. As an application, we use our methods to ascribe symbolic dynamics to perturbations of the Shilnikov homoclinic scenario and to study the dynamics of the Henon map.

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arXiv2025

Unbounded dynamics for three dimensional vector fields

Consider a three-dimensional vector field $F$ which generates a finite number of fixed points - what can we say on its unbounded dynamics? In this paper we tackle this question, and prove sufficient conditions for $F$ to have fixed points with unbounded invariant manifolds. Following that, we use these results to study the dynamics of the Genesio-Tesi system, the Belousov-Zhabotinsky reaction, and the Michelson system.

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arXiv2024

Essential dynamics in chaotic attractors

We prove that if a smooth vector field $F$ of $S^3$ generates a sufficiently complicated heteroclinic knot, the flow also generates infinitely many periodic orbits, which persist under smooth perturbations which preserve the heteroclinic knot. Consequentially, we then associate a Template with the flow dynamics - regardless of whether $F$ satisfies any hyperbolicity condition or not. In addition, inspired by the Thurston-Nielsen Classification Theorem, we also conclude topological criteria for the existence of chaotic dynamics for three-dimensional flows - which we apply to study both the R\"ossler and Lorenz attractors.

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arXiv2024

Removable dynamics in the Nose-Hoover and Moore-Spiegel Oscillators

We study the dynamics of the Nose-Hoover and Moore-Spiegel Oscillators, and in particular, their topological dynamics. We prove the dynamics of both these systems can be reduced to a flow on a solid torus, with at most a finite number of attracting periodic trajectories. As a consequence, we obtain that every periodic trajectory for the Nose-Hoover and the Moore-Spiegel Oscillators is a Torus knot.

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arXiv2023

One Dimensional Dynamics and the R\"ossler attractor

The R\"ossler system is one of the best known chaotic dynamical systems, generating a chaotic attractor which, by the numerical evidence, arises by a period-doubling route to chaos. In this paper we state and prove a topological criterion for the existence of an attractor for the R\"ossler system - and then analyze the dynamics of the non-wandering set by reducing the flow to the dynamics of a well-known one dimensional model: the Quadratic Family, $x^2+c$, $-2\leq c\leq\frac{1}{4}$.

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arXiv2023

Topological lower bounds for the R\"ossler System

The R\"ossler System is one of the best known chaotic dynamical systems, exhibiting a plethora of complex phenomena - and yet, only a few studies tackled its complexity analytically. Building on previous work by the author, in this paper we characterize the dynamical complexity for the R\"ossler System at parameter values at which the flow satisfies a certain heteroclinic condition. This will allow us to characterize the knot type of infinitely many periodic trajectories for the flow - and reduce the R\"ossler system to a simpler hyperbolic flow, capturing its essential dynamics.

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arXiv2023

Knots and Chaos in the R\"ossler System

The R\"ossler System is one of the best known chaotic dynamical systems, exhibiting a plethora of complex phenomena - and yet, only a few studies tackled its complexity analytically. In this paper we find sufficient conditions for the existence of chaotic dynamics for the R\"ossler System at some specific parameter values at which the flow satisfies a certain heteroclinic condition. This will allow us to prove the existence of infinitely many periodic trajectories for the flow, and study their bifurcations in the parameter space of the R\"ossler system.

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