Homoclinic Loop Bifurcation Research Articles

Almost Planar Homoclinic Loops in [formula omitted

In this paper we study homoclinic loops of vector fields in 3-dimensional space when the two principal eigenvalues are real of opposite sign, which we call almost planar. We are interested to have a theory for higher codimension bifurcations. Almost planar homoclinic loop bifurcations generically occur in two versions “non-twisted” and “twisted” loops. We consider high codimension homoclinic loop bifurcations under generic conditions. The generic condition forces the existence of a 2-dimensional topological invariant ring (non necessarily unique), which is a topological cylinder in the “non-twisted” case and a topological Möbius band in the “twisted” case. If the third eigenvalue is negative (resp. positive) theω-limit set (resp.α-limit set) of trajectories starting in a neighborhood of the loop is contained in the invariant ring. The dynamics is then given by fixed points or periodic points of period 2 of a 1-dimensional map which admits a nice asymptotic expansion, allowing to treat homoclinic loop bifurcations of arbitrarily high codimension in the non-twisted case.

Multiple Limit Cycle Bifurcation Surfaces and Global Families of Multiple Limit Cycles

Sufficient conditions are given for the local exitence of multiplicity- m limit cycle bifurcation surfaces, C m, of planar analytic systems depending on n parameters with n ≥ m ≥ 2. In the generic case, the surfaces C 2, C 3, and C 4 are the familiar saddle-node, cusp, and swallow-tail bifurcation surfaces, respectively. The author′s termination principle for global one-parameter families of simple limit cycles of relatively prime, planar, analytic systems is generalized to a termination principle for global one-parameter families of multiple limit cycles which implies that the boundary of a global limit cycle bifurcation surface typically consists of Hopf bifurcation surfaces and/or homoclinic (or heteroclinic) loop bifurcation surfaces.

Homoclinic Loop and Multiple Limit Cycle Bifurcation Surfaces

This paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces $\mathcal {H}$ and multiplicity-two, limit cycle bifurcation surfaces $\mathcal {C}$ for planar systems depending on two or more parameters; it determines the side of $\mathcal {H}$ or $\mathcal {C}$ on which limit cycles occur; and it shows that if $\mathcal {H}$ and $\mathcal {C}$ intersect, then typically they do so at a flat contact.

Homoclinic loop and multiple limit cycle bifurcation surfaces

This paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces H \mathcal {H} and multiplicity-two, limit cycle bifurcation surfaces C \mathcal {C} for planar systems depending on two or more parameters; it determines the side of H \mathcal {H} or C \mathcal {C} on which limit cycles occur; and it shows that if H \mathcal {H} and C \mathcal {C} intersect, then typically they do so at a flat contact.

HOMOCLINIC ORBITS AND THEIR BIFURCATIONS IN DYNAMICAL SYSTEMS WITH TWO DEGREES OF FREEDOM: A METHOD OF QUALITATIVE AND NUMERICAL ANALYSIS

An efficient method of numerical and qualitative analysis of homoclinic trajectories in Hamiltonian systems (or systems possessing a first integral of the motion) with two degrees of freedom is suggested. It allows us to reduce the problem of finding homoclinic loops as intersections of two-dimensional stable and unstable manifolds of an equilibrium point in the four-dimensional phase space to the problem of finding intersections of one-dimensional manifolds (hereafter referred to as touch-curves) on a certain two-dimensional boundary surface. The application of this algorithm in a number of problems leading to Hamiltonian dynamical systems with two degrees of freedom to the analysis of bifurcations of homoclinic loops occurring under the variation of structural parameters, is illustrated.

A Global Analysis of the Bogdanov–Takens System

This paper studies the bifurcations that occur in the Bogdanov–Takens system \[ \dot x = y,\qquad \dot y = x(x - 1) + \mu _1 y + \mu _2 xy, \] both in the phase plane ${\bf R}^2 $ and at infinity, i.e., on the Poincare sphere ${\bf S}^2 $, for all parameter values $(\mu _1 ,\mu _2 ) \in {\bf R}^2 $. For all $\mu _2 \ne 0$, a unique limit cycle bifurcates from the origin at $\mu _1 = 0$, and it expands monotonically until it intersects the saddle at $(1, 0)$ and forms a homoclinic loop at $\mu _1 = h(\mu _2 )$. The function $h(\mu _2 )$ describing the homoclinic loop bifurcation curve in ${\bf R}^2 $ is analytic for all $\mu _2 \in {\bf R},\, h(0) = 0,\,h'(0) = - 1/7$, and $h( \pm \infty ) = 0$. The only other bifurcation for this system is a saddle-saddle connection on ${\bf S}^2 $, which occurs when $\mu _1 = - 1/\mu _2 $. Even for $\mu _1 (\mu _1 \mu _2 + 1)(\mu _1 - h(\mu _2 )) \ne 0$, this system is not structurally stable on ${\bf S}^2 $ due to the presence of nonhyperbolic critical points on the equ...

A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4

In this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a Hopf bifurcation (H) of order 3 and vanishing simultaneously in a homoclinic loop bifurcation (HL) of order 3. The region with three limit cycles is a topological 3-simplex. The system is a generalization of Bogdanov's system. At the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4. This bifurcation diagram is a cone. It contains a cone on the bifurcation diagram with the three limit cycles inside a 3-simplex region, plus saddle-node and cusp bifurcations of lower order.

Saddle quantities and applications

In this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation (GHB ∗) and the practical computational aspects. For this purpose we first compare the Dulac normal form with the Joyal normal form. These forms were both used to prove the GHB ∗ theorem. But the second one is far more practical from the algorithmic point of view. We then show that the information carried by these normal forms can be computed in a much simpler way, using what we shall call dual Lyapunov constants. The coefficients of a normal form or the dual Lyapunov quantities are particular cases of what we shall call saddle quantities. We calculate the saddle quantities for quadratic systems, and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation. We also study the homoclinic loop bifurcation of order 5, appearing in a 6-parameter family close to a Hamiltonian system. To our knowledge, this is the first time that one can find a complete description of a GHB ∗ of such high order. Finally we calculate the saddle quantities for a symmetric cubic vector field, and we deduce a bound for the number of limit cycles that appear in a GHB ∗ .

Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation

We give here a planar quadratic differential system depending on two parameters, λ, δ. There is a curve in the λ-δ space corresponding to a homoclinic loop bifurcation (HLB). The bifurcation is degenerate at one point of the curve and we get a narrow tongue in which we have two limit cycles. This is the first example of such a bifurcation in planar quadratic differential systems. We propose also a model for the bifurcation diagram of a system with two limit cycles appearing at a singular point from a degenerate Hopf bifurcation, and dying in a degenerate HLB. This model shows a deep duality between degenerate Hopf bifurcations and degenerate HLBs. We give a bound for the maximal number of cycles that can appear in certain simultaneous Hopf and homoclinic loop bifurcations. We also give an example of quadratic system depending on three parameters which has at one place a degenerate Hopf bifurcation of order 3, and at another place a Hopf bifurcation of order 2 together with a HLB. We characterize the planar quadratic systems which are integrable in the neighbourhood of a homoclinic loop.

Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models.

When the traditional assumption that the incidence rate is proportional to the product of the numbers of infectives and susceptibles is dropped, the SIRS model can exhibit qualitatively different dynamical behaviors, including Hopf bifurcations, saddle-node bifurcations, and homoclinic loop bifurcations. These may be important epidemiologically in that they demonstrate the possibility of infection outbreak and collapse, or autonomous periodic coexistence of disease and host. The possible mechanisms leading to nonlinear incidence rates are discussed. Finally, a modified general criterion for supercritical or subcritical Hopf bifurcation of 2-dimensional systems is presented.