Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms

Witte, Virginie De; Govaerts, W.; Kuznetsov, Yu. А.; Meijer, Hil Gaétan Ellart

doi:10.1016/j.physd.2013.12.002

Cited by 15 publications

(16 citation statements)

References 27 publications

(58 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These equations (with unfolding parameters β 1,2 ) approximate the second iterate of the Poincaré map associated to the critical limit cycle (see [6,8]). …”

Section: Discussionmentioning

confidence: 99%

“…Fold-Neimark-Sacker bifurcation (section 2.2.1 of [8]): ξ 1 = β 1 + ξ 2 1 + s |ξ 2 | 2 , ξ 2 = (β 2 + iω 1 ) ξ 2 + (θ + iυ) ξ 1 ξ 2 + ξ 2 1 ξ 2 . …”

Section: Discussionmentioning

confidence: 99%

“…A second addition to the bifurcation structure is the fold-Neimark-Sacker bifurcation point F NS b . The normal form coefficients, computed according to the procedure described in [7,8], reveal that the unfolding corresponds to the one shown in Figure A1(c) of [8] for s = 1, θ < 0, and E > 0 and contemplates the existence of an unstable 3D torus. This torus is destroyed via a complicated global phenomena (homoclinic tangle) occurring in an exponentially small wedge in the parameter space (see [18,32,8]).…”

Section: Effects Of the F F Bubble And F Ns On The Bifurcation Structmentioning

confidence: 95%

See 2 more Smart Citations

A Degenerate 2:3 Resonant Hopf--Hopf Bifurcation as Organizing Center of the Dynamics: Numerical Semiglobal Results

Revel¹,

Alonso²,

Moiola³

2015

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

In this paper a degenerate case of a 2:3 resonant Hopf-Hopf bifurcation is studied. This codimensionfour bifurcation occurs when the frequencies of both Hopf bifurcation branches have the relation 2/3, and one of them presents the vanishing of the first Lyapunov coefficient. The bifurcation is analyzed by means of numerical two-and three-parameter bifurcation diagrams. The two-parameter bifurcation diagrams reveal the interaction of cyclic-fold, period-doubling (or flip), and NeimarkSacker bifurcations. A nontrivial bifurcation structure is detected in the main three-parameter space. It is characterized by a fold-flip (F F ) bubble interacting with curves of fold-Neimark-Sacker (F NS), generalized period-doubling (GP D), 1:2 strong resonances (R1:2), 1:1 strong resonances of periodtwo cycles (R (2) 1:1 ), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1, −1, −1), where the bifurcation curves F F , F NS, R1:2, and CH interact, are detected. A second pair of codimension-three points appears when F F interacts with GP D and R(2) 1:1 (and CH in one of the points). Finally, it is shown that this degenerate 2:3 resonant Hopf-Hopf bifurcation acts as an organizing center of the dynamics, since the structure of bifurcation curves and its singular points are unfolded by this singularity.

show abstract

“…These equations (with unfolding parameters β 1,2 ) approximate the second iterate of the Poincaré map associated to the critical limit cycle (see [6,8]). …”

Section: Discussionmentioning

confidence: 99%

“…Fold-Neimark-Sacker bifurcation (section 2.2.1 of [8]): ξ 1 = β 1 + ξ 2 1 + s |ξ 2 | 2 , ξ 2 = (β 2 + iω 1 ) ξ 2 + (θ + iυ) ξ 1 ξ 2 + ξ 2 1 ξ 2 . …”

Section: Discussionmentioning

confidence: 99%

Section: Effects Of the F F Bubble And F Ns On The Bifurcation Structmentioning

confidence: 95%

See 1 more Smart Citation

A Degenerate 2:3 Resonant Hopf--Hopf Bifurcation as Organizing Center of the Dynamics: Numerical Semiglobal Results

Revel¹,

Alonso²,

Moiola³

2015

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

show abstract

“…This type of dynamical systems occurs frequently in applications especially in the mathematical models for population biology [Jansen, 2001;Saputra et al, 2010a;De Witte et al, 2014] and for the spread of diseases [Chitnis et al, 2006;Llensa et al, 2014]. It is because in population models, if a species dies out it cannot be regenerated therefore it is always natural to have coordinate axes as the invariant manifold.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

Saputra

2015

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

We investigate a dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analyzed and yield the transcritical bifurcation as the codimension-one bifurcation while the saddle-node–transcritical interaction and the Hopf–transcritical interactions as the codimension-two bifurcations. The unfolding of this degeneracy is also analyzed and reveal global bifurcations such as homoclinic and heteroclinic bifurcations. We apply our results to a modified Lotka–Volterra model and to an infection model in HIV diseases.

show abstract

“…Compare computed and predicted periods figure(10) ; clf ; hold on ; % Plot computed periods on nsbr (1) and nsbr (2) omegas1 = arrayfun ( @ ( p ) p . period , nsbr .…”

mentioning

confidence: 99%

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Bosschaert¹,

Janssens²,

Kuznetsov³

2020

SIAM J. Appl. Dyn. Syst.

View full text Add to dashboard Cite

In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.

show abstract

Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms

Cited by 15 publications

References 27 publications

A Degenerate 2:3 Resonant Hopf--Hopf Bifurcation as Organizing Center of the Dynamics: Numerical Semiglobal Results

A Degenerate 2:3 Resonant Hopf--Hopf Bifurcation as Organizing Center of the Dynamics: Numerical Semiglobal Results

Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Contact Info

Product

Resources

About