CMUP - Dynamical Systems
https://cmup.fc.up.pt/main/seminars/dynamical-systems
en A topological route to detect chaos in two families of dynamical systems
https://cmup.fc.up.pt/main/content/topological-route-detect-chaos-two-families-dynamical-systems
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.004</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Friday, 21 September, 2018 - 11:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>The concept of chaos is widely used in the field of Dynamical Systems, and several approaches which aim to establish the presence of chaotic dynamics have been developed in the literature. At this juncture, a prototypical example comes from the geometric structure associated with the Smale’s horseshoe, cf. [4]. In recent years, several different approaches have been proposed to extend this classical geometry in a topological direction. This way, the so-called concept of “topological horseshoes” was introduced in [2].</p>
<p>The topological horseshoes along with symbolic dynamics provide a powerful tool to describe the time evolution of chaotic dynamics. In this framework, by exploiting techniques developed in [3], we investigate two families of chaotic dynamical systems. Firstly, we deal with a discrete application and prove the existence of topological horseshoes for the twisted horseshoe map [1] and its generalisation, cf. [5]. At last, we consider a continuous application and show analytically the presence of complex behaviours for a class of indefinite weight periodic boundary value problems, cf. [6].</p>
<p>References</p>
<ol><li>
[1] J. Guckenheimer, G. Oster, A. Ipaktchi, The dynamics of density dependent pop- ulation models, J. Math. Biol. 4 (1977), 101–147.
</li>
<li>
[2] J. Kennedy and J. A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc., 353 (2001), 2513–2530.
</li>
<li>
[3] A. Medio, M. Pireddu, F. Zanolin., Chaotic dynamics for maps in one and two dimensions: a geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19(10) (2009), 3283–3309.
</li>
<li>
[4] S. Smale, Finding a horseshoe on the beaches of Rio, Math. Intelligencer, 20(1) (1998), 39–44.
</li>
<li>
[5] E. Sovrano, About chaotic dynamics in the twisted horseshoe map, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26(6) (2016), 1650092, 10.
</li>
<li>
[6] E. Sovrano, How to get complex dynamics? A note on a topological approach, preprint.
</li>
</ol></div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Elisa Sovrano</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">FCUP - CMUP</div></div></section>Wed, 19 Sep 2018 14:54:42 +0000Alexandre Artur Pinho Rodrigues12878 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/topological-route-detect-chaos-two-families-dynamical-systems#commentsEquivariant Bifurcation and Ize Conjecture
https://cmup.fc.up.pt/main/content/equivariant-bifurcation-and-ize-conjecture
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.031</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Friday, 14 September, 2018 - 14:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>J Ize conjectured that for any absolutely irreducible representation of a compact Lie group G on a finite dimensional real vectorspace there exists an isotropy subgroup which has an odd dimensional fixed point space. If it were true it had immediate consequences in equivariant bifurcation. Lauterbach & Matthews showed that this is not the case. Their findings of three infinite families of finite groups were supplemented by extensive computer analysis showing a very difficult zoo of groups acting on R4. In this talk we will give a complete list of counter examples in R4. In our notation we follow the group theoretical notation by Conley & Smith.</p>
<p>We discuss the open question of infinite compact Lie groups as counter examples to the Ize conjecture. Finally we give an overview on the bifurcation scenario and an outlook on higher space dimensions.</p>
</div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Reiner Lauterbach</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">Hamburg University, Germany</div></div></section>Wed, 12 Sep 2018 13:19:23 +0000Alexandre Artur Pinho Rodrigues12877 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/equivariant-bifurcation-and-ize-conjecture#commentsDynamics on (adaptive) feedforward networks
https://cmup.fc.up.pt/main/content/dynamics-adaptive-feedforward-networks
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.030</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Thursday, 21 June, 2018 - 11:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>This talk is about describing dynamics on primitive network objects and finding conditions that allow a good "reductive" description of network dynamics. We will give a number of examples when feedback is added. The examples range from surprising synchrony, an example of the "bullwhip" effect and a remarkable layered network mixing synchrony and chaotic dynamics. Some of this work is part of a joint project with Ana Dias and Manuela Aguiar.</p>
</div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Mike Field</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">Imperial College of London, UK</div></div></section><section class="field field-name-field-seminars-homepage field-type-link-field field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker's homepage: </h2><div class="field-items"><div class="field-item even"><a href="http://math.rice.edu/~mjf8/" rel="nofollow" target="_blank" title="Speaker's homepage">http://math.rice.edu/~mjf8/</a></div></div></section>Tue, 12 Jun 2018 15:26:59 +0000Alexandre Artur Pinho Rodrigues12825 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/dynamics-adaptive-feedforward-networks#commentsOn realizing graphs as complete heteroclinic networks
https://cmup.fc.up.pt/main/content/realizing-graphs-complete-heteroclinic-networks
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.030</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Wednesday, 20 June, 2018 - 14:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>We examine the relation between a heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. In particular, we show that there are robust realizations of a large class of transitive directed graphs that are not complete (i.e. not all unstable manifolds of nodes are included) but almost complete (i.e. complete up to a set of zero measure in the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). Moreover, some of these almost complete and equable realizations have "completions", namely by adding extra nodes we can produce a larger network that is complete and may even be asymptotically stable. This is joint work with Sofia Castro (Porto) and Peter Ashwin (Exeter).</p>
</div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Alexander Lohse</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">Hamburg University</div></div></section><section class="field field-name-field-seminars-homepage field-type-link-field field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker's homepage: </h2><div class="field-items"><div class="field-item even"><a href="https://www.math.uni-hamburg.de/home/lohse/" rel="nofollow" target="_blank" title="Speaker's homepage">https://www.math.uni-hamburg.de/home/lohse/</a></div></div></section>Tue, 12 Jun 2018 15:23:46 +0000Alexandre Artur Pinho Rodrigues12823 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/realizing-graphs-complete-heteroclinic-networks#commentsRandom Lorentz gas and deterministic walks in random environments
https://cmup.fc.up.pt/main/content/random-lorentz-gas-and-deterministic-walks-random-environments
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room M031</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Friday, 15 June, 2018 - 14:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>Although one could naively expect that random Lorentz gases are easier to investigate than deterministic periodic ones, this seems not to be the case as essentially no results are available in the non periodic case. In this talk, I will present some general ideas towards studying random Lorentz gases and I will show how to apply them for a class of deterministic walks in random environments wit hone-dimensional uniformly expanding local dynamics. This is a joint work with Carlangelo Liverani.</p>
<p> </p>
</div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Romain Aimino</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">FCUP - CMUP</div></div></section>Tue, 12 Jun 2018 15:22:00 +0000Alexandre Artur Pinho Rodrigues12822 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/random-lorentz-gas-and-deterministic-walks-random-environments#comments Homogeneous coupled cell systems - unexpected symmetries and how to exploit them in bifurcation analysis
https://cmup.fc.up.pt/main/content/homogeneous-coupled-cell-systems-unexpected-symmetries-and-how-exploit-them-bifurcation
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.031</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Friday, 22 June, 2018 - 14:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>Dynamical systems with an underlying network structure are a subject of great interest as they arise frequently in applications and exhibit many staggering phenomena some of which resemble those in equivariant dynamics. We introduce a theory developed by Rink and Sanders that connects a class of network dynamical systems - namely homogeneous coupled cell systems - to equivariant dynamical systems. The symmetries, however, are generalized in the sense that they do not necessarily form a group but more general structures such as monoids or semigroups. We investigate how to exploit these symmetries in order to understand the generic bifurcation behavior of a given network. Finally we present some ideas and open questions on how to exploit representation theory of finite monoids in order to further deepen the understanding of networks.</p>
<p> </p>
</div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Sören Schwenker</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">Universität Hamburg, Germany</div></div></section>Tue, 22 May 2018 07:07:52 +0000Alexandre Artur Pinho Rodrigues12789 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/homogeneous-coupled-cell-systems-unexpected-symmetries-and-how-exploit-them-bifurcation#commentsStability of quasi-simple heteroclinic cycles
https://cmup.fc.up.pt/main/content/stability-quasi-simple-heteroclinic-cycles-0
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.031</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Friday, 4 May, 2018 - 14:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose connections are one-dimensional and contained in flow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form. Our method applies to some classes of simple heteroclinic cycles and to various heteroclinic cycles arising in population dynamics, namely non-simple cycles, as well as to heteroclinic cycles that are part of a network. We illustrate our results with a quasi-simple (non-simple) cycle in a heteroclinic network for the dynamics of the Rock-Scissors-Paper game. Using applications to price setting models, we further illustrate the contribution of the Rock-Scissors-Paper game to the understanding of cyclic dominance in two-player games.</p>
<p>References:</p>
<ol><li>
[1] L. Garrido-da-Silva and S.B.S.D. Castro (2018a) Stability of quasi-simple heteroclinic cycles. Dynamical Systems: an International Journal, <a href="https://doi.org/10.1080/14689367.2018.1445701. ">https://doi.org/10.1080/14689367.2018.1445701. </a>
</li>
<li>
[2] L. Garrido-da-Silva and S.B.S.D. Castro (2018b) Cyclic dominance in a two-person Rock-Scissors-Paper game. arXiv:1607.08748.
</li>
</ol></div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Liliana Garrido da Silva</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">FCUP -- CMUP</div></div></section>Sun, 29 Apr 2018 16:03:14 +0000Alexandre Artur Pinho Rodrigues12741 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/stability-quasi-simple-heteroclinic-cycles-0#commentsThe Conley index and its applications to the study of surface flows
https://cmup.fc.up.pt/main/content/conley-index-and-its-applications-study-surface-flows
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.005</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Friday, 27 April, 2018 - 11:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>In this seminar we shall introduce the Conley index of an isolated invarant set of a flow on a locally compact metric space. The Conley index is a homotopical tool which encapsulates dynamical information near the isolated invariant set. The definition of this invariant involves the use of some external objects, namely isolating blocks (or, more generally, index pairs). We will give a way to compute this index in "intrinsic terms" for flows defined on surfaces. To do this we will deepen into the structure of the unstable manifold of an isolated invariant set. This description will allow us to give a complete classification of the possible Conley indices of an isolated invariant continuum in a surface and to derive some interesting dynamical consequences.</p>
<p> </p>
</div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Hector Barge</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">Universidad Politécnica de Madrid (Spain)</div></div></section>Wed, 25 Apr 2018 11:09:20 +0000Alexandre Artur Pinho Rodrigues12734 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/conley-index-and-its-applications-study-surface-flows#commentsEstabilização de ciclos heterodimensionais
https://cmup.fc.up.pt/main/content/estabiliza%C3%A7%C3%A3o-de-ciclos-heterodimensionais
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.031</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Friday, 20 April, 2018 - 14:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>Os célebres resultados de S.Newhouse ([N]) mostram que a bifurcação de uma tangência homoclínica asociada a uma sela numa superfície gera tangências homoclínicas robustas (isto é, tangências homoclínicas que persistem por pequenas perturbações) associadas a um conjunto hiperbólico especial chamado ferradura espessa. Além disso, a continuação (hiperbólica) da sela inicial está contida nesse conjunto hiperbólico. Neste caso diz-se que a tangência pode ser estabilizada, ou seja, perturbações arbitrariamente pequenas da tangência homoclínica associada a uma sela geram ferraduras espessas (com tangências homoclínicas) contendo a continuação da sela inicial.</p>
<p>No contexto de ciclos (heterodimensionais), Ch.Bonatti, L.J.Díaz e S. Kiriki, em [BD,BDK], mostraram que os ciclos associados a um par de selas (em dimensão ao menos três) podem ser estabilizados, isto é, perturbações arbitrariamente pequenas do ciclo inicial (associado a selas) geram ciclos robustos contendo as continuações das selas iniciais.</p>
<p> </p>
<p>Nos resultados citados, a regularidade da topologia desempenha um papel fundamental: a construção em [N] aplica em topologias de classe pelo menos C^2 e os resultado [BD,BDK] aplicam-se na C1-topologia. Limitações técnicas impedem-nos de adaptar diretamente os trabalhos [N] ao caso C^1 nem [BD,BDK] ao caso C^r com r>1. Neste seminário discutiremos o problema de estabilização de ciclos em dimensão 3 na C^r-topologia com r>1.</p>
<p> </p>
<p>[N]: S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Etudes Sci., 50, 101–151, (1979).</p>
<p>[BD]: Ch. Bonatti and L. J. DÍaz, Robust heterodimensional cycles and C1-generic dynamics, Journal of the Inst. of Math. Jussieu, 7(3) (2008), 469–525.</p>
<p>[BDK]: C. Bonatti, L. J. Díaz, and S. Kiriki, Stabilization of heterodimensional cycles, Nonlinearity, 25 (2012), p. 931.</p>
</div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Sebastian Perez</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">FCUP -- CMUP</div></div></section>Wed, 18 Apr 2018 08:07:09 +0000Alexandre Artur Pinho Rodrigues12723 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/estabiliza%C3%A7%C3%A3o-de-ciclos-heterodimensionais#commentsStrange attractors near a homoclinic cycle to a bifocus
https://cmup.fc.up.pt/main/content/strange-attractors-near-homoclinic-cycle-bifocus
<div class="field field-name-field-seminars-place field-type-text field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even">Room FC1.031</div></div></div><div class="field field-name-field-date-events field-type-datetime field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><span class="date-display-single">Friday, 6 April, 2018 - 14:30</span></div></div></div><div class="field field-name-field-seminars-area field-type-taxonomy-term-reference field-label-hidden view-mode-rss"><ul class="field-items"><li class="field-item even"><a href="/main/seminars/dynamical-systems">Dynamical Systems</a></li></ul></div><div class="field field-name-body field-type-text-with-summary field-label-hidden view-mode-rss"><div class="field-items"><div class="field-item even"><p>In this seminar, we explore the chaotic set near a homoclinic cycle to a hyperbolic bifocus at which the vector field has negative divergence. If the invariant manifolds of the bifocus satisfy a non-degeneracy condition, a sequence of hyperbolic suspended horseshoes arises near the cycle, with one expanding and two contracting directions.</p>
<p>We extend previous results on the field and we show that, when the cycle is broken, there are parameters for which the first return map to a given cross section exhibits homoclinic tangencies associated to a dissipative saddle periodic point. These tangencies can be slightly modified in order to satisfy the Tatjer conditions for a generalized tangency of codimension two. This configuration may be seen the organizing center, by which one can obtain strange attractors and infinitely many sinks.</p>
<p>Therefore, the existence of a homoclinic cycle associated to a bifocus may be considered as a criterion for four-dimensional flows to be $C^1$-approximated by other flows exhibiting strange attractors.</p>
</div></div></div><section class="field field-name-field-seminars-speaker field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Speaker: </h2><div class="field-items"><div class="field-item even">Alexandre Rodrigues</div></div></section><section class="field field-name-field-speaker-institution field-type-text field-label-inline clearfix view-mode-rss"><h2 class="field-label">Institution: </h2><div class="field-items"><div class="field-item even">CMUP - FCUP</div></div></section>Sat, 31 Mar 2018 12:09:06 +0000Alexandre Artur Pinho Rodrigues12673 at https://cmup.fc.up.pt/mainhttps://cmup.fc.up.pt/main/content/strange-attractors-near-homoclinic-cycle-bifocus#commentsError | CMUP
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