114 research outputs found
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
Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems
Systems that are not smooth can undergo bifurcations that are forbidden in
smooth systems. We review some of the phenomena that can occur for
piecewise-smooth, continuous maps and flows when a fixed point or an
equilibrium collides with a surface on which the system is not smooth. Much of
our understanding of these cases relies on a reduction to piecewise linearity
near the border-collision. We also review a number of codimension-two
bifurcations in which nonlinearity is important.Comment: pdfLaTeX, 9 figure
Computation of periodic solution bifurcations in ODEs using bordered systems
We consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous ODEs, namely the fold, the period-doubling (or flip) bifurcation, and the torus (or Neimark–Sacker) bifurcation. In the fold and flip cases we append one scalar equation to the standard periodic BVP that defines the periodic solution; in the torus case four scalar equations are appended. Evaluation of these scalar equations and their derivatives requires the solution of linear BVPs, whose sparsity structure (after discretization) is identical to that of the linearization of the periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software AUTO and COLSYS
Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
We study the dynamics of the five-parameter quadratic family of
volume-preserving diffeomorphisms of R^3. This family is the unfolded normal
form for a bifurcation of a fixed point with a triple-one multiplier and also
is the general form of a quadratic three-dimensional map with a quadratic
inverse. Much of the nontrivial dynamics of this map occurs when its two fixed
points are saddle-foci with intersecting two-dimensional stable and unstable
manifolds that bound a spherical ``vortex-bubble''. We show that this occurs
near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at
least in its normal form, an elliptic invariant circle. We develop a simple
algorithm to accurately compute these elliptic invariant circles and their
longitudinal and transverse rotation numbers and use it to study their
bifurcations, classifying them by the resonances between the rotation numbers.
In particular, rational values of the longitudinal rotation number are shown to
give rise to a string of pearls that creates multiple copies of the original
spherical structure for an iterate of the map.Comment: 53 pages, 29 figure
From A.Tondl’s Dutch contacts to Neimark-Sacker-bifurcation
We present a description of the many contacts of A. Tondl with Dutch scientists involving nonlinear dynamics models for mechanics. One of the topics is Neimark-Sacker bifurcation that leads to the presence of families of quasi-periodic solutions that are geometrically organised and visualised in tori. A new model in the spirit of A. Tondl, containing interaction of self-excited and parametrically excited oscillators is analysed to find this bifurcation and quasi-periodic solutions. The analysis using averaging in combination with numerical bifurcation tools Matcont and Auto produces a picture of rich dynamical phenomena with several surprises among which a special quasi-periodic solution produced by the averaged equation
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