Quiver Representations and Dimension Reduction in Dynamical Systems

We study emergent oscillatory behavior in networks of diffusively coupled nonlinear ordinary differential equations. Starting from a situation where each isolated node possesses a globally attracting equilibrium point, we give, for an arbitrary network configuration, general conditions for the existence of the diffusive coupling of a homogeneous strength which makes the network dynamics chaotic. The method is based on the theory of local bifurcations we develop for diffusively coupled networks. We, in particular, introduce the class of the so-called versatile network configurations and prove that the Taylor coefficients of the reduction to the center manifold for any versatile network can take any given value.

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“…We also point out that symmetry is often instrumental in explaining and predicting anomalous behavior in network dynamical systems [1,14,15].…”

Section: Informal Statement Of Main Resultsmentioning

confidence: 91%

Chaotic Behavior in Diffusively Coupled Systems

Nijholt

Pereira

Queiroz

et al. 2023

Commun. Math. Phys.

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“…Note that these symmetries are parts of larger structures, namely the subnetwork quiver and the quotient quiver. For more details, see [19]. (3.4).…”

Section: Proof the Bifurcation Assumption (B) Impliesmentioning

confidence: 99%

Amplified steady state bifurcations in feedforward networks

Gracht

Nijholt²,

Rink

2022

Nonlinearity

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We investigate bifurcations in feedforward coupled cell networks. Feedforward structure (the absence of feedback) can be defined by a partial order on the cells. We use this property to study generic one-parameter steady state bifurcations for such networks. Branching solutions and their asymptotics are described in terms of Taylor coefficients of the internal dynamics. They can be determined via an algorithm that only exploits the network structure. Similar to previous results on feedforward chains, we observe amplifications of the growth rates of steady state branches induced by the feedforward structure. However, contrary to these earlier results, as the interaction scenarios can be more complicated in general feedforward networks, different branching patterns and different amplifications can occur for different regions in the space of Taylor coefficients.

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“…The branching solutions in Theorem 3.7 are the same as the ones described in Proposition 5.1 in [29] for layered feedforward networks (compare to Remark 2. 19) and in Proposition 5.7 of [2] investigating feedforward structure of transitive components. Here we simply extend these results by the explicit computation of the leading coefficients.…”

Section: The Critical Cells Are Maximalmentioning

confidence: 99%

“…Note that these symmetries are parts of larger structures, namely the subnetwork quiver and the quotient quiver. For more details on this, see [19].…”

Section: Branches Of Steady States For the Entire Networkmentioning

confidence: 99%

Amplified steady state bifurcations in feedforward networks

von der Gracht,

Nijholt,

Rink

2021

Preprint

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We investigate bifurcations in feedforward coupled cell networks. Feedforward structure (the absence of feedback loops) can be defined by a partial order on the cells. We use this property to study generic one-parameter steady state bifurcations for such networks. Branching solutions and their asymptotics are described in terms of Taylor coefficients of the internal dynamics. They can be determined via an algorithm that only exploits the network structure. Similar to previous results on feedforward chains, we observe amplifications of the growth rates of steady state branches induced by the feedforward structure. However, contrary to these earlier results, as the interaction scenarios can be more complicated in general feedforward networks, different branching patterns and different amplifications can occur for different regions in the space of Taylor coefficients.

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Quiver Representations and Dimension Reduction in Dynamical Systems

Cited by 11 publications

References 23 publications

Chaotic Behavior in Diffusively Coupled Systems

Chaotic Behavior in Diffusively Coupled Systems

Amplified steady state bifurcations in feedforward networks

Amplified steady state bifurcations in feedforward networks

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