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Heteroclinic cycles and networks
Stability of cycles within networks
Coupled cell systems
Local and global bifurcations
Singularities and strange attractors
Organizing Committee:
Manuela Aguiar (CMUP and FEP.UP)
Isabel Labouriau (CMUP and FCUP)
Alexandre Rodrigues (CMUP and ISEG)
Participants (updated June 2023):
Sofia Castro
Isabel Labouriau
Manuela Aguiar
Alexandre Rodrigues
Santiago Ibáñez
Alexander Lohse
Liliana Garrido-da-Silva
Telmo Peixe
Haibo Ruan
Ana Paula Dias
Soeren von der Gracht
Chris Bick
João Maurício de Carvalho
Ana Margarida Ferreira
Official Program
Mini-courses and tutorials
02/Nov (14h30m; Thursday): Sofia Castro (FEP, CMUP)
Subject: Heteroclinic dynamics
03/Nov (14h30m, Friday): Manuela Aguiar (FEP, CMUP)
Subject: Coupled Cell Networks: basic definitions and results
06/Nov (12h00m, Monday): Liliana Garrido-da-Silva (FCUP, CMUP)
Subject: Stability of cycles and networks
07/Nov (14h30m, Tuesday): Ana Paula Dias (FCUP, CMUP)
Subject: Network constraints on bifurcations of coupled cell systems
09/Nov (14h30m, Thursday): Santiago Ibañez (University of Oviedo)
Subject: Singularities: unfolding and coupling
10/Nov (14h30m, Friday): Alexandre Rodrigues (ISEG, CMUP)
Subject: Chaos in homoclinic bifurcations
Talks
13/Nov (14h30m, Monday): Alexander Lohse (Hamburg University)
14/Nov (14h30m, Tuesday): Chris Bick (Amsterdam University)
15/Nov (16h30m, Wednesday): Soeren von der Gracht (Paderborn University)
16/Nov (14h30m, Thursday): Haibo Ruan (Hamburg University of Technology)
17/Nov (14h30m, Friday): Telmo Peixe (REM, CEMAPRE, ISEG -- Lisbon University)
Titles and Abstracts
Titles and Abstracts
First part: Tutorials and Mini-courses
First part: Tutorials and Mini-courses
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Day: 02nd of November, 14h30
Day: 02nd of November, 14h30
Speaker: Sofia Castro (CMUP, FEP - University of Porto)
Speaker: Sofia Castro (CMUP, FEP - University of Porto)
Duration: 50min + 50min
Duration: 50min + 50min
Room: FC1.006
Title: Heteroclinic dynamics
Title: Heteroclinic dynamics
Abstract:
Abstract:
In ordinary differential equations, a saddle-saddle connection is generically not robust. However, when either symmetry or extinction hyperplanes are present these give rise to flow-invariant spaces, where this connection becomes a saddle-sink connection that is robust. A sequence of connections between consecutive equilibria is called a heteroclinic cycle. A heteroclinic network is a connected union of heteroclinic cycles.
When a cycle is part of a heteroclinic network it cannot be asymptotically stable. It can nevertheless exhibit some stability that may make the cycle visible in experiments and simulations. I shall describe several intermediate notions of stability and ways to determine them.
This mini-course will be more pedagogical than a standard scientific seminar.
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Day: 03rd of November, 14h30
Day: 03rd of November, 14h30
Speaker: Manuela Aguiar (CMUP, FEP - University of Porto)
Speaker: Manuela Aguiar (CMUP, FEP - University of Porto)
Duration: 1h30m
Duration: 1h30m
Room: FC1.031
Title: Coupled Cell Networks: basic definitions and results
Title: Coupled Cell Networks: basic definitions and results
Abstract:
Abstract:
In this course we will introduce the basic definitions of the coupled cell networks formalism. In particular, we will define different kinds of networks, in what concerns their topology and the existence or not of weights on the connections, and the coupled systems that are admissible by them. One of the key aspects of this formalism is the capacity of deducing dynamical properties of the admissible systems of a network based only on the topology of the network. One of those properties is the existence of subspaces of the phase space that are flow invariant by all the admissible systems. This is an important property that is observed in many models in different real world areas and that can impact greatly the dynamics, as we will see. We will also see how it iis possible to get the lattice of the synchrony subspaces of a given network.
In this course we will introduce the basic definitions of the coupled cell networks formalism. In particular, we will define different kinds of networks, in what concerns their topology and the existence or not of weights on the connections, and the coupled systems that are admissible by them. One of the key aspects of this formalism is the capacity of deducing dynamical properties of the admissible systems of a network based only on the topology of the network. One of those properties is the existence of subspaces of the phase space that are flow invariant by all the admissible systems. This is an important property that is observed in many models in different real world areas and that can impact greatly the dynamics, as we will see. We will also see how it iis possible to get the lattice of the synchrony subspaces of a given network.
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Day: 06th of November, 12h00
Day: 06th of November, 12h00
Speaker: Liliana Garrido-da-Silva (CMUP)
Speaker: Liliana Garrido-da-Silva (CMUP)
Duration: 1h
Duration: 1h
Room: FC1.005
Title: Stability of cycles and networks
Title: Stability of cycles and networks
Abstract:
Abstract:
The stability of heteroclinic trajectories within a heteroclinic cycle or network can be quantified by means of the local stability index. We derive explicit expressions for the local stability indices for a general class of robust heteroclinic cycles called quasi-simple heteroclinic cycles. A heteroclinic cycle is quasi-simple if its heteroclinic connections are one-dimensional and contained in flow-invariant spaces of equal dimensions. We ensure that the dynamics between two connected equilibria is encoded in a transition matrix whose entries only depends on the eigenvalues of the linearisation at the outgoing equilibrium. The local stability index is calculated by taking the rows of the (products of) suitable transition matrices. We illustrate our method with quasi-simple cycles arising in models from game theory, population dynamics and neuroscience. Part of this course is based on joint work with Sofia Castro (FEP, CMUP).
The stability of heteroclinic trajectories within a heteroclinic cycle or network can be quantified by means of the local stability index. We derive explicit expressions for the local stability indices for a general class of robust heteroclinic cycles called quasi-simple heteroclinic cycles. A heteroclinic cycle is quasi-simple if its heteroclinic connections are one-dimensional and contained in flow-invariant spaces of equal dimensions. We ensure that the dynamics between two connected equilibria is encoded in a transition matrix whose entries only depends on the eigenvalues of the linearisation at the outgoing equilibrium. The local stability index is calculated by taking the rows of the (products of) suitable transition matrices. We illustrate our method with quasi-simple cycles arising in models from game theory, population dynamics and neuroscience. Part of this course is based on joint work with Sofia Castro (FEP, CMUP).
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Day: 07th of November, 14h30
Day: 07th of November, 14h30
Speaker: Ana Paula Dias (CMUP, FCUP - University of Porto)
Speaker: Ana Paula Dias (CMUP, FCUP - University of Porto)
Duration: 45min + 45min
Duration: 45min + 45min
Room: FC1.031
Title: Network constraints on bifurcations of coupled cell systems
Title: Network constraints on bifurcations of coupled cell systems
Abstract:
Abstract:
We address some interesting network questions that have been posed in the last years. One concerns when two distinct networks are equivalent from the point of view of the sets of network admissible dynamical systems; the answer to this question effects the possibility of networks classification and we illustrate all that with small excitatory-inhibitory networks. In the second part of this talk, we intend to illustrate the application of Liapunov-Schmidt reduction to the study of steady-state bifurcations in network admissible dynamical systems. Finally, we present the class of Hopf-steady-state networks aiming to illustrate how network constrains on bifurcations of network admissible dynamical systems. Part of the talk is based on joint works with Aguiar (Porto), Mokhtari (Vrije Universiteit Amsterdam, Holland) and Stewart (Warwick, UK).
We address some interesting network questions that have been posed in the last years. One concerns when two distinct networks are equivalent from the point of view of the sets of network admissible dynamical systems; the answer to this question effects the possibility of networks classification and we illustrate all that with small excitatory-inhibitory networks. In the second part of this talk, we intend to illustrate the application of Liapunov-Schmidt reduction to the study of steady-state bifurcations in network admissible dynamical systems. Finally, we present the class of Hopf-steady-state networks aiming to illustrate how network constrains on bifurcations of network admissible dynamical systems. Part of the talk is based on joint works with Aguiar (Porto), Mokhtari (Vrije Universiteit Amsterdam, Holland) and Stewart (Warwick, UK).
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Day: 09th of November, 14h30
Day: 09th of November, 14h30
Speaker: Santiago Ibáñez (University of Oviedo)
Speaker: Santiago Ibáñez (University of Oviedo)
Duration: 50min + 50min
Duration: 50min + 50min
Room: FC1.006
Title: Singularities: unfolding and coupling
Title: Singularities: unfolding and coupling
Abstract:
Abstract:
When two or more dynamical systems interact, new behaviors emerge. For example, synchronization (or desynchronization) phenomena may occur. Also, as Alan Turing observed, oscillations can occur where behavior was previously stationary, or even chaotic dynamics where evolution was previously predictable. These are some of the topics we will address in the context of couplings.
When two or more dynamical systems interact, new behaviors emerge. For example, synchronization (or desynchronization) phenomena may occur. Also, as Alan Turing observed, oscillations can occur where behavior was previously stationary, or even chaotic dynamics where evolution was previously predictable. These are some of the topics we will address in the context of couplings.
Our tool will be the study of singularities (local bifurcations) that occur in coupled systems. We will see how the study of their unfoldings (the atlas of dynamics that arise around the singularity) sometimes allows us to determine properties of synchronisation/desynchronization or the existence of chaotic dynamics.
We will first approach singularities and their unfoldings from a very general perspective and then focus on three particularly interesting cases: Hopf-Zero and Hopf-Hopf singularities, and nilpotent singularities. Moreover, our dynamics will be governed by families of vector fields, i.e., we will refer to singularities of vector fields (points at which the vector fields are zero) and to couplings of systems of differential equations.
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Day: 10th of November, 14h30
Speaker: Alexandre Rodrigues (ISEG and CMUP)
Duration: 1h30m
Room: FC1.031
Title: Chaos in homoclinic bifurcations
Abstract:
Proving the existence of a homo- or heteroclinic connection in a given vector field is not an easy task. Santiago Ibanez showed how certain types of non-hyperbolic singularities (nilpotent singularities) generically unfold fields with homo- and heteroclinic connections. In this talk, we describe some global bifurcations (homo and heteroclinic) in autonomous differential equations, and their impact on the geometry of the associated dynamics. Special emphasis will be given to the emergence of chaos.
Results of this talk may be found in:
P. G. Barrientos, S. Ibáñez, A. A. Rodrigues, J. A. Rodriguez, Emergence of Chaotic Dynamics from Singularities, 32th Brazilian Mathematics Colloquium, Instituto Nacional de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, 2019. xi+200 pp. ISBN: 978-85-244-0430-6, 2019
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Second part: Talks and Seminars
Second part: Talks and Seminars
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Day: 13th of November, 14h30
Day: 13th of November, 14h30
Speaker: Alexander Lohse (Hamburg University)
Speaker: Alexander Lohse (Hamburg University)
Duration: 1h
Duration: 1h
Room: FC1.031
Title: Switching dynamics
Title: Switching dynamics
Abstract:
Abstract:
In every heteroclinic network there exists at least one node (equilibrium) with more than one outgoing connection towards another node. Trajectories in a neighbourhood of the network may therefore "choose" which of these connections to follow. For a given network, it is possible to analyze systematically for which sequences of heteroclinic connections there are initial conditions near the network such that the solution follows exactly the prescribed sequence of connections. This helps to understand the long-term dynamics of a system given through an ODE. Studying such questions has led to the notion of switching dynamics and a wide range of dynamic behaviour has been unveiled subsequently, related among other things to the eigenvalues of the linearization of the vector field at the nodes.
In this talk we introduce different levels of switching and give a basic idea of conditions under which they may or may not occur. We start by looking at a simple example and finish with some questions which currently draw considerable attention.
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Day: 14th of November, 14h30
Day: 14th of November, 14h30
Speaker: Christian Bick (Amsterdam University)
Speaker: Christian Bick (Amsterdam University)
Duration: 1h
Duration: 1h
Room: FC1.006
Title: Heteroclinic Dynamics between Localized Synchrony Patterns and Forced Symmetry Breaking
Title: Heteroclinic Dynamics between Localized Synchrony Patterns and Forced Symmetry Breaking
Abstract:
Abstract:
Networks of identical oscillators can exhibit synchrony patterns where synchrony is localized in part of the network: A subset
of oscillators are synchronized while the rest is incoherent. While the focus has predominantly been on stable patterns, we show that for specific (nonpairwise) coupling there can also be heteroclinic cycles and networks that allow dynamical transitions between distinct localized synchrony patterns. This analysis relies on the specific choice of coupling that give rise to continuous symmetries such that the patterns are actually equilibria relative to this symmetry action. Since we cannot expect these symmetries to be present, we analyze how the synchrony patterns deform as the continuous symmetries are broken. Using numerical continuation, we identify how the synchrony patterns bifurcate in the perturbed system.
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Day: 15th of November, 16h30
Day: 15th of November, 16h30
Speaker: Soeren von der Gracht (Paderborn University)
Speaker: Soeren von der Gracht (Paderborn University)
Duration: 1h30m
Duration: 1h30m
Room: FC1.006
Title: Strange symmetries and strange bifurcations in network dynamical systems
Title: Strange symmetries and strange bifurcations in network dynamical systems
Abstract:
Abstract:
Many dynamical systems in fields such as neuroscience (the workings of the brain), systems biology (metabolic systems) and robotics (robot swarms) exhibit the structure of a network: they consist of nodes (neurons, proteins, robots) with connections between them. It usually does not suffice to understand the nature of the individual nodes to deduce the behavior of the network, as the specific interaction structure of a network can produce remarkable dynamics. Prominent examples include synchronization (e.g., the simultaneous firing of neurons) and highly complex branching behavior in bifurcations, phenomena that are not found in dynamical systems without the structure of a network.
Network dynamical systems are not well understood mathematically, which makes it hard to quantify and control their behavior. The reason is that most of the established machinery of dynamical systems theory fails to distinguish between networks and general dynamical systems. Thus, we need mathematical tools that are tailor-made for network problems. Several techniques have been proposed recently, and they strikingly have one thing in common: they exploit the algebraic nature of networks.
In this talk, I will give an overview over some recent results regarding the question which dynamical behavior and generic bifurcations are dictated by the network structure of a system. In particular, I will illustrate how structural and algebraic properties culminate in symmetries of the governing equations and how these can be exploited for (partial) answers. This includes classical symmetries but also more exotic concepts such as monoid and quiver representations.
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Day: 16th of November, 14h30
Day: 16th of November, 14h30
Speaker: Haibo Ruan (Hamburg University of Technology)
Speaker: Haibo Ruan (Hamburg University of Technology)
Duration: 1h
Duration: 1h
Room: FC1.006
Title: Synchrony Patterns in Gene Regulatory Networks
Title: Synchrony Patterns in Gene Regulatory Networks
Abstract:
Abstract:
Motivated by studying synchronization mechanisms in gene regulatory networks (GRNs) and their relation to evolutionary events such as genetic duplication and genetic redundancy, we consider two mathematical dynamical models of GRNs. We obtain results on robust synchronization on these dynamical models inspired by the existing theoretical results in the coupled cell network formalisms. We also explore the concepts of quotient networks and network lifting in the context of GRNs which are related to the process of gene duplication and the phenomenon of subfunctionalization as an outcome of functional divergence.
Motivated by studying synchronization mechanisms in gene regulatory networks (GRNs) and their relation to evolutionary events such as genetic duplication and genetic redundancy, we consider two mathematical dynamical models of GRNs. We obtain results on robust synchronization on these dynamical models inspired by the existing theoretical results in the coupled cell network formalisms. We also explore the concepts of quotient networks and network lifting in the context of GRNs which are related to the process of gene duplication and the phenomenon of subfunctionalization as an outcome of functional divergence.
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Day: 17th of November, 14h30
Day: 17th of November, 14h30
Speaker: Telmo Peixe (REM, CEMAPRE, ISEG -- University of Lisbon)
Speaker: Telmo Peixe (REM, CEMAPRE, ISEG -- University of Lisbon)
Duration: 50min + 50min
Duration: 50min + 50min
Room: FC1.031
Title: Dynamics along the heteroclinic network of polymatrix replicators
Title: Dynamics along the heteroclinic network of polymatrix replicators
Abstract:
Abstract:
The polymatrix replicator is a system of ordinary differential equations that can be used to model the time evolution of behavioural strategies of in- dividuals in a stratified population. The flow of these systems evolve on a prism (polytope) given by the product of simplices. In this talk we present a new method to analyse the asymptotic dynamics of a flow on a polytope along its edge-vertex heteroclinic network. For this purpose we will explore the dynamics of some examples of polymatrix replicators. A significant part of this work is joint work with Alexandre Rodrigues (ISEG, CMUP).
The polymatrix replicator is a system of ordinary differential equations that can be used to model the time evolution of behavioural strategies of in- dividuals in a stratified population. The flow of these systems evolve on a prism (polytope) given by the product of simplices. In this talk we present a new method to analyse the asymptotic dynamics of a flow on a polytope along its edge-vertex heteroclinic network. For this purpose we will explore the dynamics of some examples of polymatrix replicators. A significant part of this work is joint work with Alexandre Rodrigues (ISEG, CMUP).
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Picture by Juliane Oliveira, Sofia Castro and Isabel Labouriau
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