Lorenz attractors in unfoldings of homoclinic-flip bifurcations

Golmakani, Ali; Homburg, Ale Jan

doi:10.1080/14689367.2010.503186

Cited by 17 publications

(8 citation statements)

References 30 publications

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“…Hence, this term can be viewed as a time rescaling that does not alter the dynamics of the vector fields; therefore, it can be omitted. As mentioned before,w = 0 is an invariant plane for (11), (12) and (13) that represents infinity. After substitution of the corresponding polynomials and simplification of the expressions, we setw = 0 in (11), (12) and (13), which leads to the following three vector fields that represent the dynamics of system (2) at infinity in the corresponding charts:…”

Section: A3 Analytical Study Of Infinitymentioning

confidence: 89%

“…We remark that the conditions for the homoclinic flip bifurcation of a hyperbolic equilibrium have been studied for the non-hyperbolic case, namely, for the case of a transcritical bifurcation [24]; the authors show that the non-hyperbolic equilibrium gives rise to new heteroclinic orbits and that its unfolding is different from the hyperbolic case. Reference [11] explores the creation of a Lorenz-like attractor in homoclinic loop configurations that exhibit homoclinic flip bifurcations; this happens when two homoclinic orbits connect to the same equilibrium, which in [11] is studied by looking at systems with reflectional symmetry.…”

Section: Homoclinic Flip Bifurcationsmentioning

confidence: 99%

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Saddle Invariant Objects and Their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

Giraldo¹,

Krauskopf²,

Osinga³

2017

SIAM J. Appl. Dyn. Syst.

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When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way, provided the corresponding genericity conditions. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or non-orientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case B, which is characterized by the fact that the codimension-two point gives rise to an additional homoclinic bifurcation, namely, a two-homoclinic orbit. To explain how the global manifolds organize phase space, we consider Sandstede's three-dimensional vector field model, which features inclination and orbit flip bifurcations. We compute global invariant manifolds and their intersection sets with a suitable sphere, by means of continuation of suitable two-point boundary problems, to understand their role as separatrices of basins of attracting periodic orbits. We show representative images in phase space and on the sphere, such that we can identify topological properties of the manifolds in the different regions of parameter space and at the homoclinic bifurcations involved. We find heteroclinic orbits between saddle periodic orbits and equilibria, which give rise to regions of infinitely many heteroclinic orbits. Additional equilibria exist in Sandstede's model and we compactify phase space to capture how equilibria may emerge from or escape to infinity. We present images of these bifurcation diagrams, where we outline different configurations of equilibria close to homoclinic flip bifurcations of case B; furthermore, we characterize the dynamics of Sandstede's model at infinity.

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Section: A3 Analytical Study Of Infinitymentioning

confidence: 89%

Section: Homoclinic Flip Bifurcationsmentioning

confidence: 99%

Saddle Invariant Objects and Their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

Giraldo¹,

Krauskopf²,

Osinga³

2017

SIAM J. Appl. Dyn. Syst.

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“…For instance, orbit flip points can be found in the Hindmarsh-Rose model, which is a simplified polynomial model of a certain class of neuron cells [47]. Orbit flip bifurcations also play a role in the creation of Lorenz-like attractors of systems with reflectional symmetry [19]. Inclination flips, on the other hand, organize the emergence of mixed-mode oscillations in a Van der Pol-Duffing model [29], and they are associated with the excitable behaviour of reaction-diffusion systems subjected to nonlocal coupling [5].…”

Section: Introductionmentioning

confidence: 99%

Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation

Aguirre¹,

Krauskopf²,

Osinga³

2013

SIAM J. Appl. Dyn. Syst.

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Homoclinic bifurcations are important phenomena that cause global rearrangements of the dynamics in phase space, including changes to basins of attractions and the generation of chaotic dynamics. We consider here a homoclinic (or connecting) orbit that converges in both forward and backward time to a saddle equilibrium of a three-dimensional vector field. We assume that the saddle is such that the eigenvalues of its Jacobian are real. If such a homoclinic orbit is broken by varying a suitable parameter then, generically, a single periodic orbit Γ bifurcates. We consider the case that the saddle quantity of the equilibrium is negative so that Γ is an attractor (rather than of saddle type). At the moment of bifurcation the two-dimensional stable manifold of the saddle, when followed along the homoclinic orbit, may form either an orientable or nonorientable surface, and one speaks of an orientable or a nonorientable homoclinic bifurcation. A change of orientability occurs at two kinds of codimension-two homoclinic bifurcations, namely, an inclination flip and an orbit flip. The stable manifold of the saddle point is neither orientable nor nonorientable at either of these bifurcations. In this paper we study how the stable manifold of the saddle organizes the phase space globally near these homoclinic bifurcations. To this end, we consider a model vector field due to Sandstede, in which the origin 0 is a saddle point that undergoes the respective homoclinic bifurcations for certain choices of the parameters. We compute its global stable manifold W s (0) via the continuation of suitable orbit segments to determine how it changes through the bifurcation in question. More specifically, we render W s (0) as a two-dimensional surface in the three-dimensional phase space, and also consider its intersection set with a suitable sphere. We first investigate the transition through the orientable and nonorientable codimension-one homoclinic bifurcations (with negative saddle quantity); in particular, we show how the basin of attraction of the bifurcating periodic orbit Γ is created in each case. We then study the global invariant manifold W s (0) near the transition between these two cases as given by an inclination flip and an orbit flip bifurcation. More specifically, we present two-parameter bifurcation diagrams of the two flip bifurcations with representative images, in phase space and on the sphere, of W s (0) in relation to other relevant invariant objects. In this way, we identify the topological properties of W s (0) in open regions of parameter space and at the bifurcations involved.

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“…The theorems differ depending on the local monotonicity of the two functions in a neighborhood of the border point. The assumptions are given mainly on qualitative properties and the proofs are new, depending on topological properties instead of more analytical ones which have recently been published, among which it is worth mentioning [Labarca & Moreira, 2001Golmakani & Homburg, 2011;Labarca & Moreira, 2010;Winckler, 2011;Labarca & Vásquez, 2012].…”

Section: Introductionmentioning

confidence: 99%

Codimension-2 Border Collision, Bifurcations in One-Dimensional, Discontinuous Piecewise Smooth Maps

Gardini

Аврутин

Sushko

2014

Int. J. Bifurcation Chaos

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We consider a two-parametric family of one-dimensional piecewise smooth maps with one discontinuity point. The bifurcation structures in a parameter plane of the map are investigated, related to codimension-2 bifurcation points defined by the intersections of two border collision bifurcation curves. We describe the case of the collision of two stable cycles of any period and any symbolic sequences. For this case, we prove that the local monotonicity of the functions constituting the first return map defined in a neighborhood of the border point at the parameter values related to the codimension-2 bifurcation point determines, under suitable conditions, the kind of bifurcation structure originating from this point; this can be either a period adding structure, or a period incrementing structure, or simply associated with the coupling of colliding cycles.

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Lorenz attractors in unfoldings of homoclinic-flip bifurcations

Cited by 17 publications

References 30 publications

Saddle Invariant Objects and Their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

Saddle Invariant Objects and Their Global Manifolds in a Neighborhood of a Homoclinic Flip Bifurcation of Case B

Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation

Codimension-2 Border Collision, Bifurcations in One-Dimensional, Discontinuous Piecewise Smooth Maps

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