Singularity theory and forced symmetry breaking in equations

Furter, Jacques E.; Ruas, María Aparecida Soares; Sitta, Angela Maria

doi:10.5565/publmat_54210_01

Publicacions Matemàtiques

2010

DOI: 10.5565/publmat_54210_01

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Singularity theory and forced symmetry breaking in equations

Jacques E. Furter¹,

María Aparecida Soares Ruas²,

Angela Maria Sitta³

Abstract: A theory of bifurcation equivalence for forced symmetry breaking bifurcation problems is developed. We classify (O(2), 1) problems of corank 2 of low codimension and discuss examples of bifurcation problems leading to such symmetry breaking.

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Cited by 2 publications

References 29 publications

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Forced Symmetry-Breaking Bifurcations in Networks with Disordered Parameters

Palacios,

Sorrentino,

Nazerian

et al. 2023

Int. J. Bifurcation Chaos

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Emergent behavior in complex networks can be predicted and analyzed via the mechanism of spontaneous symmetry-breaking bifurcation, in which solutions of related bifurcation problems lose symmetry as some parameters are varied, even though the equations that such solutions satisfy retain the full symmetry of the system. A less common mechanism is that of forced symmetry-breaking, in which either a bifurcation problem has symmetry on both the state variables and the parameters, or one where the equations have less symmetry when a certain parameter is varied. In this manuscript, it is shown that in certain networks with parameter mismatches the governing equations remain unchanged when the group of symmetries acts on both the state variables and the parameter space. Based on this observation we study the existence and stability of collective patterns in symmetric networks with parameters mismatches from the point of view of forced symmetry-breaking bifurcations. Treating the parameters as state variables, we perform center manifold reductions, which allow us to understand how the disorder in parameters affects the bifurcation points as well as the stability properties of the ensuing patterns. Theoretical results are validated with numerical simulations.

show abstract

Forced Symmetry-Breaking Bifurcations in Networks with Disordered Parameters

Palacios,

Sorrentino,

Nazerian

et al. 2023

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

show abstract

Disorder-Induced Dynamics in Complex Networks

Palacios,

In,

Amani

2024

Int. J. Bifurcation Chaos

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Disorder in parameters appears to influence the collective behavior of complex adaptive networks in ways that might seem unconventional. For instance, heterogeneities may, unexpectedly, lead to enhanced regions of existence of stable synchronization states. This behavior is unexpected because synchronization appears, generically, in symmetric networks with homogeneous components. Related works have, however, misidentified cases where disorder seems to play a critical role in enhancing synchronization, where it is actually not the case. Thus, in order to clarify the role of disorder in adaptive networks, we use normal forms to study, mathematically, when and how the presence of disorder can facilitate the emergence of collective patterns. We employ parameter symmetry breaking to study the interplay between disorder and the underlying bifurcations that determine the conditions for the existence and stability of collective behavior. This work provides a rigorous justification for a certain barycentric condition to be imposed on the heterogeneity of the parameters while studying the synchronization state. Theoretical results are accompanied by numerical simulations, which help clarify incorrect claims of disorder purportedly enhancing synchronization states.

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Singularity theory and forced symmetry breaking in equations

Abstract: A theory of bifurcation equivalence for forced symmetry breaking bifurcation problems is developed. We classify (O(2), 1) problems of corank 2 of low codimension and discuss examples of bifurcation problems leading to such symmetry breaking.

Cited by 2 publications

References 29 publications

Forced Symmetry-Breaking Bifurcations in Networks with Disordered Parameters

Forced Symmetry-Breaking Bifurcations in Networks with Disordered Parameters

Disorder-Induced Dynamics in Complex Networks

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