Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

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

Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

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

Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

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

Periodic Center Manifolds and Normal Forms for DDEs in the Light of Suns and Stars

Bifurcation theory has been very successful in the study of qualitative changes in nonlinear dynamical systems. An important tool of this theory is the existence of a center manifold near nonhyperbolic equilibria and limit cycles or homoclinic orbits. The existence has already been proven for many kinds of different systems, but not fully for limit cycles in delay differential equations (DDEs). In this paper, we prove the existence of a smooth finite-dimensional periodic center manifold near a nonhyperbolic cycle in DDEs and the existence of a special coordinate system on this manifold. This allows us to describe the local dynamics on the center manifold in terms of the standard normal forms. These results are based on the rigorous functional analytic perturbation framework for dual semigroups, also called sun-star calculus

Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

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