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Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

Annalisa Iuorio, Christian Kuehn, Peter Szmolyan
2018 Discrete and Continuous Dynamical Systems. Series S  
Based on the geometric analysis we construct an initial periodic orbit to start numerical continuation of periodic orbits with respect to the key parameters.  ...  The same functional with more general boundary conditions has been treated by Vainchtein et al [48] .  ...  CK and PS also acknowledge partial support of the European Commission (EC/REA) via a Marie-Curie International Reintegration Grant (MC-IRG).  ... 
doi:10.3934/dcdss.2020073 fatcat:zkdeicyccncyjc5x2tdepddohq

Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem [article]

Annalisa Iuorio, Christian Kuehn, Peter Szmolyan
2017 arXiv   pre-print
We investigate a singularly perturbed, non-convex variational problem arising in materials science with a combination of geometrical and numerical methods.  ...  Based on the geometric analysis we construct an initial periodic orbit to start numerical continuation of periodic orbits with respect to the key parameters.  ...  CK and PS also acknowledge partial support of the European Commission (EC/REA) via a Marie-Curie International Reintegration Grant (MC-IRG).  ... 
arXiv:1701.08153v1 fatcat:ofsdztglpnb6hppzrxsauxhum4

Homoclinic and Heteroclinic Bifurcations in Vector Fields [chapter]

Ale Jan Homburg, Björn Sandstede
2010 Handbook of Dynamical Systems  
Furthermore, v j (µ) is a multiple of the eigenvector of f u (p(µ), µ) associated with ν j (µ) for j = s, u. In addition, there is a constant > 0 so that and analogously for h u (t; µ).  ...  Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of  ...  We conclude this section with a few remarks on heteroclinic bifurcations for planar ODEs.  ... 
doi:10.1016/s1874-575x(10)00316-4 fatcat:fadh4mnuw5erlpcikmoiqoa6oe

Heteroclinic dynamics in the nonlocal parametrically driven nonlinear Schrödinger equation

M Higuera, J Porter, E Knobloch
2002 Physica D : Non-linear phenomena  
The origin of these transitions is explained using a careful study of a two-mode Galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic  ...  As the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics.  ...  generate a mean flow.  ... 
doi:10.1016/s0167-2789(01)00368-2 fatcat:bfsalwqnpjhtxojf57f4mz2qea

The onset of chaos in vortex sheet flow

2002 Journal of Fluid Mechanics  
The resulting section displays the generic features of a chaotic Hamiltonian system, resonance bands and a heteroclinic tangle, and these features are well-correlated with the irregular features in the  ...  A spectral analysis is performed to determine the fundamental oscillation frequency and this is used to construct a Poincaré section of the vortex sheet flow.  ...  In summary, the Poincaré section of the vortex sheet flow displays the generic features of a chaotic Hamiltonian system: resonance bands and a heteroclinic tangle.  ... 
doi:10.1017/s0022112001007066 fatcat:p2qaqi3rlves7ecujm7pscbct4

Energy Conservation, Singular Orbits, and FPGA Implementation of Two New Hamiltonian Chaotic Systems

Enzeng Dong, Guanghan Liu, Zenghui Wang, Zengqiang Chen
2020 Complexity  
In this paper, two Hamiltonian conservative chaotic systems (HCCSs) are constructed based on the 4D Euler equations and a proposed construction method.  ...  Since the conservative chaotic system (CCS) has no general attractors, conservative chaotic flows are more suitable for the chaos-based secure communication than the chaotic attractors.  ...  with a heteroclinic orbit connecting two of the equilibria [37] . ey also found that the rupture of this cycle would separate the chaotic attractors.  ... 
doi:10.1155/2020/8693157 fatcat:tmj5nokce5acjnjjsbqlpxpo7e

Time-reversal symmetry in dynamical systems: A survey

Jeroen S.W. Lamb, John A.G. Roberts
1998 Physica D : Non-linear phenomena  
We follow with a survey of the state of the art on the theory of reversible dynamical systems, including results on symmetric periodic orbits, local bifurcation theory, homoclinic orbits, and renormalization  ...  In this paper we survey the topic of time-reversal symmetry in dynamical systems. We begin with a brief discussion of the position of time-reversal symmetry in physics.  ...  Appendix A has its roots in the bibliographies of Sevryuk [Sevryuk, 1991b] and Roberts and Quispel [Roberts and Quispel, 1992] , and was updated and extended with the use of Mathematical Reviews (Math  ... 
doi:10.1016/s0167-2789(97)00199-1 fatcat:bbt5u3jhibe6tmis3q73nkksw4

Mixmaster: fact and belief

J Mark Heinzle, Claes Uggla
2009 Classical and quantum gravity  
We consider the dynamics towards the initial singularity of Bianchi type IX vacuum and orthogonal perfect fluid models with a linear equation of state.  ...  The evidence for the conjectures is based on a study of the stochastic properties of this map in conjunction with dynamical systems techniques.  ...  We gratefully acknowledge the hospitality of the Mittag-Leffler Institute, where part of this work was completed. CU is supported by the Swedish Research Council.  ... 
doi:10.1088/0264-9381/26/7/075016 fatcat:xex3frc5njhjxmbcnj6cfxtfj4

Spiralling dynamics near heteroclinic networks

Alexandre A.P. Rodrigues, Isabel S. Labouriau
2014 Physica D : Non-linear phenomena  
We construct explicitly a two parameter family of vector fields on the three-dimensional sphere ^3, whose flow has a spiralling attractor containing the following: two hyperbolic equilibria, heteroclinic  ...  The spiralling set unfolds a heteroclinic network between two symmetric saddle-foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols  ...  Acknowledgements: The authors would like to express their gratitude to Manuela Aguiar (University of Porto) for helpful discussions at the beginning of this work. Also special thanks  ... 
doi:10.1016/j.physd.2013.10.012 fatcat:djidyyj6efgonefaflg565s2fa

Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

Svetoslav G. Nikolov, Vassil M. Vassilev
2021 Mathematics  
Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator.  ...  In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs-Rössler prototype-4 system, was investigated.  ...  The generic homoclinic orbit with complex principal eigenvalues (bifocal homoclinic orbit) has been studied by numerous researchers [25, [27] [28] [29] .  ... 
doi:10.3390/math9040352 fatcat:ejspds73kzbzfeentvp2l3s5my

Connectivity and design of planar global attractors of Sturm type. II: Connection graphs

Bernold Fiedler, Carlos Rocha
2008 Journal of Differential Equations  
The unique heteroclinic orbits between adjacent Morse levels define a plane graph C f which we call the connection graph.  ...  The planar Sturm attractor consists of equilibria of Morse index 0, 1, or 2, and their heteroclinic connecting orbits.  ...  Acknowledgments For bearing endless discussions with limitless enduring patience we are much indebted to Stefan Liebscher.  ... 
doi:10.1016/j.jde.2007.09.015 fatcat:uz2ddnfklnc6phz3wz4xz35dty

Computational Challenges of Nonlinear Systems [chapter]

Laurette S. Tuckerman
2020 Regularity and Complexity in Dynamical Systems  
In order of increasing complexity, we describe methods for calculating steady states and bifurcation diagrams, linear stability and Floquet analysis, and heteroclinic orbits.  ...  These are illustrated by computations for Rayleigh-Bénard convection in a cylindrical geometry, the Faraday instability of a fluid layer, the flow past a cylinder and over a square cavity, flow in a cylindrical  ...  Steady states and bifurcation diagrams: cylindrical convection Dynamical systems can be organized around objects of increasing complexity: steady states, periodic orbits, and then tori, heteroclinic orbits  ... 
doi:10.1007/978-3-030-44992-6_11 fatcat:l4vqzpmodbhblfev7qssnc53m4

Central configurations

Richard Moeckel
2014 Scholarpedia  
We now take the final step, combining homoclinic and heteroclinic orbits of the same energy value to generate what is called a homoclinic-heteroclinic chain of orbits, which connect asymptotically the  ...  Along the orbit, these two vectors generate a family of planes which are sent from one into another by the variational flow.  ... 
doi:10.4249/scholarpedia.10667 fatcat:3ccrmgvrxrerdkhejh4utxduyq

Spatially localized structures in dissipative systems: open problems

E Knobloch
2008 Nonlinearity  
The recent resurgence in interest in these structures has led to significant advances in our understanding of the origin and properties of these states, and these in turn suggest new questions, both general  ...  optics, fluid flow, surface catalysis, neurobiology and many more.  ...  I am grateful to J Burke for helpful comments, and to A Alonso, P Assemat, O Batiste, A Bergeon, J Burke, I Mercader and A Yochelis for collaboration on questions related to the topic of this paper.  ... 
doi:10.1088/0951-7715/21/4/t02 fatcat:wy7ygzvnrbbivaq4ghc5siqbem

Hamiltonian Perturbation Theory (and Transition to Chaos) [chapter]

Henk W. Broer, Heinz Hanßmann
2009 Encyclopedia of Complexity and Systems Science  
But also perturbations that destroy the Hamiltonian character are important, be it to study the effect of a small amount of friction, or to further the theory of dissipative systems themselves which surprisingly  ...  The fundamental problem of mechanics is to study Hamiltonian systems that are small perturbations of integrable systems.  ...  In that case two small parameters play a role, namely Planck's constant as well as the distance away from integrability.  ... 
doi:10.1007/978-0-387-30440-3_267 fatcat:q6gmt6kqzvdfxazukukg5s2z64
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