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Trans-heteroclinic bifurcation : a novel type of catastrophic shift
Sardanyés, Josep (Centre de Recerca Matemàtica)
Martínez Barchino, Regina (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Simó, Carles (Barcelona Graduate School of Mathematics)

Date: 2018
Abstract: Global and local bifurcations are extremely important since they govern the transitions between different qualitative regimes in dynamical systems. These transitions or tipping points, which are ubiquitous in nature, can be smooth or catastrophic. Smooth transitions involve a continuous change in the steady state of the system until the bifurcation value is crossed, giving place to a second-order phase transition. Catastrophic transitions involve a discontinuity of the steady state at the bifurcation value, giving place to first-order phase transitions. Examples of catastrophic shifts can be found in ecosystems, climate, economic or social systems. Here we report a new type of global bifurcation responsible for a catastrophic shift. This bifurcation, identified in a family of quasi-species equations and named as trans-heteroclinic bifurcation, involves an exchange of stability between two distant and heteroclinically connected fixed points. Since the two fixed points interchange the stability without colliding, a catastrophic shift takes place. We provide an exhaustive description of this new bifurcation, also detailing the structure of the replication-mutation matrix of the quasi-species equation giving place to this bifurcation. A perturbation analysis is provided around the bifurcation value. At this value the heteroclinic connection is replaced by a line of fixed points in the quasi-species model. But it is shown that, if the replication-mutation matrix satisfies suitable conditions, then, under a small perturbation, the exchange of heteroclinic connections is preserved, except on a tiny range around the bifurcation value whose size is of the order of magnitude of the perturbation. The results presented here can help to understand better novel mechanisms behind catastrophic shifts and contribute to a finer identification of such transitions in theoretical models in evolutionary biology and other dynamical systems.
Note: Número d'acord de subvenció MINECO/MTM2016-80116-P
Note: Número d'acord de subvenció MINECO/MDM-2014-0445
Note: Número d'acord de subvenció MINECO/MTM2013-41168-P
Note: Número d'acord de subvenció AGAUR/2014/SGR-1145
Rights: Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original. Creative Commons
Language: Anglès.
Document: article ; recerca ; publishedVersion
Subject: Global bifurcations ; Nonlinear dynamics ; Quasi-species equations ; Catastrophic shifts ; Trans-heteroclinic bifurcation ; Applied mathematics
Published in: Royal Society Open Science, Vol. 5 (Jan. 2018) , art. 171304, ISSN 2054-5703

PMID: 29410837
DOI: 10.1098/rsos.171304


12 p, 847.0 KB

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Articles > Research articles
Articles > Published articles

 Record created 2018-03-06, last modified 2020-04-25



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