Global Dynamics of the Kummer–Schwarz Differential Equation

Llibre, Jaume; Vidal, Cláudio

doi:10.1007/s00009-013-0299-4

Cited by 5 publications

(10 citation statements)

References 6 publications

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“…This procedure can be generalized to higher dimensions, the best-known case is the compactification to the Poincaré-disk of polynomial vector fields on R 2 [8,12]. We are interested here in R 3 , which has been studied for certain models [25,26,27,44] through the use of coordinate charts.…”

Section: A Poincaré Compactificationmentioning

confidence: 99%

“…The three-dimensional vector fields in these projections contain subsets of the equator S 2 R 4 that correspond to invariant planes. Hence, the problem of studying the dynamics at infinity can be simplified to a study of two-dimensional vector fields [25,26,27,44]. For ease of notation, we use the variablesx,ỹ,z andw interchangeably in the different charts.…”

Section: A3 Analytical Study Of Infinitymentioning

confidence: 99%

“…When α = 0, condition (24) implies that u(0) lies in the plane normal to Γ at v Γ (0); this is a necessary condition for v Γ (0) to achieve the minimal distance of u(0) to Γ. Condition (25) defines the radius of the tubular section, that is, it ensures that u(0) lies on T d Γ . The sequence of steps to follow in Auto is: 1.…”

Section: B Boundary Value Problem Formulationsmentioning

confidence: 99%

“…2. Extend the system with the 2PBVP-formulation (21) to (25), and define a first solution u = u(0) = u(1) ∈ F δ with T = 0.…”

Section: B Boundary Value Problem Formulationsmentioning

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: A Poincaré Compactificationmentioning

confidence: 99%

Section: A3 Analytical Study Of Infinitymentioning

confidence: 99%

Section: B Boundary Value Problem Formulationsmentioning

confidence: 99%

“…2. Extend the system with the 2PBVP-formulation (21) to (25), and define a first solution u = u(0) = u(1) ∈ F δ with T = 0.…”

Section: B Boundary Value Problem Formulationsmentioning

confidence: 99%

See 2 more Smart Citations

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.

View full text Add to dashboard Cite

show abstract

“…For a definition of Poincaré disc and Poincaré sphere see [6] and [5,15], respectively. When we say the singularities at the endpoints of the axes, we mean the singular points which are on the boundary of the Poincaré disc or of the Poincaré sphere at the endpoints of the coordinate axes.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%