arXiv subjects

Alexander Lohse

Publications and source records attributed to Alexander Lohse.

15 recordsLinked to original sources

arXiv2026

Design of Hierarchical Excitable Networks

We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ (the lower level), together with another digraph $\mathbf{\Gamma}$ on $N$ vertices (the top level). The dynamic realizations of $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\mathbf{\Gamma}$ correspond to transitions between these different patterns. In our construction, the connections given through $\mathbf{\Gamma}$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $\delta$-neighborhood of the first set contains an initial condition with $\omega$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.

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arXiv2025

Complete heteroclinic networks derived from graphs consisting of two cycles

We address the question how a given connection structure (directed graph) can be realised as a heteroclinic network that is complete in the sense that it contains all unstable manifolds of its equilibria. For a directed graph consisting of two cycles we provide a constructive method to achieve this: (i) enlarge the graph by adding some edges and (ii) apply the simplex method to obtain a network in phase space. Depending on the length of the cycles we derive the minimal number of required new edges. In the resulting network each added edge leads to a positive transverse eigenvalue at the respective equilibrium. We discuss the total number of such positive eigenvalues in an individual cycle and some implications for the stability of this cycle.

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arXiv2024

How many points converge to a heteroclinic network in an aperiodic way?

Homoclinic and heteroclinic connections can form cycles and networks in phase space, which organize global phenomena in dynamical systems. On the one hand, stability notions for (omni)cycles give insight into how many initial conditions approach the network along a single given (omni)cycle. On the other hand, the term switching is used to describe situations where there are trajectories that follow any possible sequence of heteroclinic connections along the network. Here we give a notion of asymptotic stability for general sequences along a network of homoclinic and heteroclinic connections. We show that there cannot be uncountably many aperiodic sequences that attract a set with nontrivial measure. Finally, we discuss examples where one may or may not expect aperiodic convergence towards a network and conclude with some open questions.

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arXiv2023

Arbitrarily large heteroclinic networks in fixed low-dimensional state space

We consider heteroclinic networks between $n \in \mathbb{N}$ nodes where the only connections are those linking each node to its two subsequent neighbouring ones. Using a construction method where all nodes are placed in a single one-dimensional space and the connections lie in coordinate planes, we show that it is possible to robustly realise these networks in $\mathbb{R}^6$ for any number of nodes $n$ using a polynomial vector field. This bound on the space dimension (while the number of nodes in the network goes to $\infty$) is a novel phenomenon and a step towards more efficient realisation methods for given connection structures in terms of the required number of space dimensions. We briefly discuss some stability properties of the generated heteroclinic objects.

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arXiv2022

Stability Indices of Non-Hyperbolic Equilibria in Two-Dimensional Systems of ODEs

We consider families of systems of two-dimensional ordinary differential equations with the origin $0$ as a non-hyperbolic equilibrium. For any number $s \in (-\infty, +\infty)$ we show that it is possible to choose a parameter in these equations such that the stability index $σ(0)$ is precisely $σ(0)=s$. In contrast to that, for a hyperbolic equilibrium $x$ it is known that either $σ(x)=-\infty$ or $σ(x)=+\infty$. Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability indices.

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arXiv2022

A hybrid heteroclinic cycle

Using a vector field in $\mathbb{R}^4$, we provide an example of a robust heteroclinic cycle between two equilibria that displays a mix of features exhibited by well-known types of low-dimensional heteroclinic structures, including simple, quasi-simple and pseudo-simple cycles. Our cycle consists of two equilibria on one coordinate axis and two connections. One of the connections is one-dimensional while the other is two-dimensional. We compare our heteroclinic cycle to others in the literature that are similar in architecture, and illustrate how the standard methods used to analyse those cycles fail to provide sufficient information on the attraction properties of our example. The instability of two subcycles contained in invariant three-dimensional subspaces seems to indicate that our cycle is generically completely unstable. Although this cycle is one of the simplest possible and exists in low-dimension, the complete study of the stability of our cycle by using the standard techniques for return map reduction is not possible given the hybrid nature of the return map.

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arXiv2019

Moduli of stability for heteroclinic cycles of periodic solutions

We consider $C^2$ vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic behavior and from the asymptotic properties of its orbits we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. As expected, this set contains the periods of the orbits involved in the cycle, a combination of their angular speeds, the rates of expansion and contraction in linearizing neighborhoods of them, besides information regarding the transition maps and the transition times between these neighborhoods. We conclude with an application of this result to a class of cycles obtained by the lifting of an example of R. Bowen.

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arXiv2019

Almost complete and equable heteroclinic networks

Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the relation between the heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. We consider realizations of a large class of transitive digraphs as robust heteroclinic networks and show that although robust realizations are typically not complete (i.e. not all unstable manifolds of nodes are part of the network), they can be almost complete (i.e. complete up to a set of zero measure within the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). We show there are almost complete and equable realizations that can be closed by adding a number of extra nodes and connections. We discuss some examples and describe a sense in which an equable almost complete network embedding is an optimal description of stochastically perturbed motion on the network.

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arXiv2018

Heteroclinic Dynamics of Localized Frequency Synchrony: Stability of Heteroclinic Cycles and Networks

In the first part of this paper, we showed that three coupled populations of identical phase oscillators give rise to heteroclinic cycles between invariant sets where populations show distinct frequencies. Here, we now give explicit stability results for these heteroclinic cycles for populations consisting of two oscillators each. In systems with four coupled phase oscillator populations, different heteroclinic cycles can form a heteroclinic network. While such networks cannot be asymptotically stable, the local attraction properties of each cycle in the network can be quantified by stability indices. We calculate these stability indices in terms of the coupling parameters between oscillator populations. Hence, our results elucidate how oscillator coupling influences sequential transitions along a heteroclinic network where individual oscillator populations switch sequentially between a high and a low frequency regime; such dynamics appear relevant for the functionality of neural oscillators.

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arXiv2018

Simple heteroclinic networks in ${\mathbb R}^4$

We classify simple heteroclinic networks for a $Γ$-equivariant system in ${\mathbb R}^4$ with finite $Γ\subset {\rm O}(4)$, proceeding as follows: we define a graph associated with a given $Γ\subset {\rm O}(n)$ and identify all so-called simple graphs associated with subgroups of ${\rm O}(4)$. Then, knowing the graph associated with a given $Γ$, we determine the types of heteroclinic networks that the group admits. Our study is restricted to networks that are maximal in the sense that they have the highest possible number of connections -- any non-maximal network can then be derived by deleting one or more connections. Finally, for networks of type A, i.e., admitted by $Γ\subset {\rm SO}(4)$, we give necessary and sufficient conditions for fragmentary and essential asymptotic stability. (For other simple heteroclinic networks the conditions for stability are known.) The results are illustrated by a numerical example of a simple heteroclinic network that involves two subcycles that can be essentially asymptotically stable simultaneously.

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arXiv2018

Pseudo-simple heteroclinic cycles in $R^4$

We study pseudo-simple heteroclinic cycles for a $Γ$-equivariant system in $R^4$ with finite $Γ\subset O(4)$, and their nearby dynamics. In particular, in a first step towards a full classification - analogous to that which exists already for the class of simple cycles - we identify all finite subgroups of $O(4)$ admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups $Γ\subset O(4)$, $Γ\not\subset SO(4)$, admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for $Γ\subset SO(4)$ pseudo-simple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.

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arXiv2017

Boundary crisis for degenerate singular cycles

The term boundary crisis refers to the destruction or creation of a chaotic attractor when parameters vary. The locus of a boundary crisis may contain regions of positive Lebesgue measure marking the transition from regular dynamics to the chaotic regime. This article investigates the dynamics occurring near a heteroclinic cycle involving a hyperbolic equilibrium point E and a hyperbolic periodic solution P, such that the connection from E to P is of codimension one and the connection from P to E occurs at a quadratic tangency (also of codimension one). We study these cycles as organizing centers of two-parameter bifurcation scenarios and, depending on properties of the transition maps, we find different types of shift dynamics that appear near the cycle. Breaking one or both of the connections we further explore the bifurcation diagrams previously begun by other authors. In particular, we identify the region of crisis near the cycle, by giving information on multipulse homoclinic solutions to E and P as well as multipulse heteroclinic tangencies from P to E, and bifurcating periodic solutions, giving partial answers to the problems (Q1)-(Q3) of E. Knobloch (2008), Spatially localised structures in dissipative systems: open problems, Nonlinearity, 21, 45-60. Throughout our analysis, we focus on the case where E has real eigenvalues and P has positive Floquet multipliers.

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arXiv2016

Construction of heteroclinic networks in $\mathbb{R}^4$

We study heteroclinic networks in $\mathbb{R}^4$, made of a certain type of simple robust heteroclinic cycle. In simple cycles all the connections are of saddle-sink type in two-dimensional fixed-point spaces. We show that there exist only very few ways to join such cycles together in a network and provide the list of all possible such networks in $\mathbb{R}^4$. The networks involving simple heteroclinic cycles of type A are new in the literature and we describe the stability of the cycles in these networks: while the geometry of type A and type B networks is very similar, stability distinguishes them clearly.

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arXiv2016

Switching in heteroclinic networks

We study the dynamics near heteroclinic networks for which all eigenvalues of the linearization at the equilibria are real. A common connection and an assumption on the geometry of its incoming and outgoing directions exclude even the weakest forms of switching (i.e. along this connection). The form of the global transition maps, and thus the type of the heteroclinic cycle, plays a crucial role in this. We look at two examples in $\mathbb{R}^5$, the House and Bowtie networks, to illustrate complex dynamics that may occur when either of these conditions is broken. For the House network, there is switching along the common connection, while for the Bowtie network we find switching along a cycle.

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arXiv2014

Stability in simple heteroclinic networks in $\mathbb{R}^4$

We describe all heteroclinic networks in $\mathbb{R}^4$ made of simple heteroclinic cycles of types $B$ or $C$, with at least one common connecting trajectory. For networks made of cycles of type $B$, we study the stability of the cycles that make up the network as well as the stability of the network. We show that even when none of the cycles has strong stability properties the network as a whole may be quite stable. We prove, and provide illustrative examples of, the fact that the stability of the network does not depend {\em a priori} uniquely on the stability of the individual cycles.

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