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

Aguirre, Pablo; Krauskopf, Bernd; Osinga, Hinke M.

doi:10.1137/130912542

Cited by 28 publications

(65 citation statements)

References 40 publications

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“…The theoretical results describe the unfoldings of the dynamics locally in a small tubular neighbourhood of the homoclinic orbit. A "more global" approach, which relies on numerical computations, has been used in [1] to understand how the global manifolds re-arrange phase space for the simplest case A. Already for this case an extra bifurcating branch of heteroclinic folds was found that had previously not been identified.…”

Section: Introductionmentioning

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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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: Introductionmentioning

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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“…Finally, notice that this work is also useful in that it shows the scope of what can be achieved in population dynamics with a combination of bifurcation theory, advanced numerical techniques for dynamical systems, and invariant manifold analysis; see also Aguirre et al as other examples of this modern blend of diverse tools. For instance, similar techniques are currently being applied by one of these authors to study a similar model with double Allee effect and a Leslie‐Gower type of approach for the conversion rate of captured preys into births of new predators …”

Section: Discussionmentioning

confidence: 94%

“…1,2 Typically, the stable manifold W s (x * ) is the mathematical object that separates the basins of 2 different attractors. Hence, to characterise any particular basin of attraction, one needs to find the corresponding stable manifolds of saddle objects in phase space that form its basin boundary; see, for instance, Aguirre et al [3][4][5] In the case of population models, this analysis allows one to give specific conditions for extinction and survival.…”

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confidence: 99%

“…6 Moreover, stable manifolds can undergo critical rearrangements under parameter variation. 1,2,4,5 These reconfigurations of stable manifolds can produce dramatic transitions in the basins they separate. 3 Hence, a combination of stable manifold analysis and bifurcation theory emerges as a necessary strategy to understand how the phase space is partitioned by different basins of attraction.…”

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confidence: 99%

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Allee thresholds and basins of attraction in a predation model with double Allee effect

Julio

Aguirre

2018

Math Methods in App Sciences

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We investigate the nature of Allee thresholds and basins of attraction in a predation model with double Allee effect in the prey and a competition behaviour in the predator. From a mathematical perspective, this implies to find and characterise the corresponding basin boundaries in phase space. This is typically a major challenge since the objects that act as boundaries between 2 different basins are invariant manifolds of the system, which may also undergo topological changes at bifurcations. For this goal, we make an extensive use of analytical tools from dynamical systems theory and numerical bifurcation analysis and determine the full bifurcation diagram. Local bifurcations include saddle‐node, transcritical and Hopf bifurcations, while global phenomena include homoclinic bifurcations, heteroclinic connections, and heteroclinic cycles. We identify the Allee threshold to be either a limit cycle, a homoclinic orbit, or the stable manifold of an equilibrium. This strategy based on bifurcation and invariant manifold analysis allows us to identify the mathematical mechanisms that produce rearrangements of separatrices in phase space. In this way, we give a full geometrical explanation of how the Allee threshold and basins of attraction undergo critical transitions. This approach is complemented with a study of the dynamics near infinity. In this way, we determine the conditions such that the basins of attraction are bounded or unbounded sets in phase space. All in all, these results allow us to show a complete description of phase portraits, extinction thresholds, and basins of attraction of our model under variation of parameters.

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“…Inclination flip bifurcations have been linked to emerging stability windows of stable equilibria and periodic orbits in the Shimizu-Morioka system [67]. These numerous inclination flip bifurcations, of which there exist several types [40], and their role for the organization of nearby chaotic dynamics and stability windows can be studied in the spirit of recent investigations [3,30]. This is an interesting direction for future work.…”

Section: Further First Foliation Tangencies In Thementioning

confidence: 97%

Finding First Foliation Tangencies in the Lorenz System

Creaser¹,

Krauskopf²,

Osinga³

2017

SIAM J. Appl. Dyn. Syst.

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Abstract. Classical studies of chaos in the well-known Lorenz system are based on reduction to the onedimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic Lorenz attractor. This reduction requires that the stable and unstable foliations in a particular Poincaré section are transverse locally near the chaotic Lorenz attractor. We study when this so-called foliation condition fails for the first time and the classic Lorenz attractor becomes a quasi-attractor. This transition is characterized by the creation of tangencies between the stable and unstable foliations and the appearance of hooked horseshoes in the Poincaré return map. We consider how the threedimensional phase space is organized by the global invariant manifolds of saddle equilibria and saddle periodic orbits-before and after the loss of the foliation condition. We compute these global objects as families of orbit segments, which are found by setting up a suitable two-point boundary value problem (BVP). We then formulate a multi-segment BVP to find the first tangency between the stable foliation and the intersection curves in the Poincaré section of the two-dimensional unstable manifold of a periodic orbit. It is a distinct advantage of our BVP setup that we are able to detect and readily continue the locus of first foliation tangency in any plane of two parameters as part of the overall bifurcation diagram. Our computations show that the region of existence of the classic Lorenz attractor is bounded in each parameter plane. It forms a slanted (unbounded) cone in the three-parameter space with a curve of terminal-point, or T-point, bifurcations on the locus of first foliation tangency; we identify the tip of this cone as a codimension-three T-point-Hopf bifurcation point, where the curve of T-point bifurcations meets a surface of Hopf bifurcation. Moreover, we are able to find other first foliation tangencies for larger values of the parameters that are associated with additional T-point bifurcations: each tangency adds an extra twist to the central region of the quasi-attractor.

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

Cited by 28 publications

References 40 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

Allee thresholds and basins of attraction in a predation model with double Allee effect

Finding First Foliation Tangencies in the Lorenz System

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