Narrow Results

This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,…) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the Rössler equation.

We describe a method for the design of feedback stabilization control laws for nonlinear systems using the theory of normal forms and the results from optimal control theory. We show that the resulting controllers can provide a larger region of stability than local linear controllers designed to perform the same task.

According to [Yu, 1999], at most two terms remain in the amplitude equation of the normal form of a continuous system, expressed in polar coordinates, with a Hopf or Generalized Hopf singularity, if we (only) apply specific nonlinear transformation to the conventional normal form; but, at least one remains in the phase equation. In this paper we show that, using a particular nonlinear scaling, these terms in the phase equation can be eliminated, which simplifies the (diffeomorphic) normal form given by [Yu, 1999]. Besides, we have also treated the Neimark–Sacker and Generalized Neimark–Sacker bifurcations, the corresponding discrete cases of the Hopf and Generalized Hopf, obtaining results which are similar to the continuous ones.

The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Hénon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincaré map are computed using variational equations to find the normal form coefficients.

Simple computational formulas are derived for the two-, three-, and four-order coefficients of the smooth normal form on the center manifold at the Bogdanov–Takens (nonsemisimple double-zero) bifurcation for n-dimensional systems with arbitrary n ≥ 2. These formulas are equally suitable for both symbolic and numerical evaluation and allow one to classify all codim 3 Bogdanov–Takens bifurcations in generic multidimensional ODEs. They are also applicable to systems with symmetries. We perform no preliminary linear transformations but use only critical (generalized) eigenvectors of the linearization matrix and its transpose. The derivation combines the approximation of the center manifold with the normalization on it. Three known models are used as test examples to demonstrate advantages of the method.

An explicit, computationally efficient, recursive formula is presented for computing the normal form and center manifold of general n-dimensional systems associated with Hopf bifurcation. Maple program is developed based on the analytical formulas, and shown to be computationally efficient, using two examples.

We face the problem of characterizing the periodic cases in parametric families of rational diffeomorphisms of 𝕂^k, where 𝕂 is ℝ or ℂ, having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two- and three-dimensional classes of polynomial or rational maps. In particular, we find the global periodic cases for several Lyness-type recurrences.

The field of dynamical systems had been revolutionized by the seminal work of Leonid Shil'nikov. As a tribute to his genius we analyze in this paper the response of dynamical systems to systematic variations of a control parameter in time, using a normal form approach. Explicit expressions of the normal forms and of their parameter dependences are derived for a class of systems possessing multiple steady-states associated to collective choices between several options in group-living organisms, giving rise to bifurcations of the pitchfork and of the limit point type. Depending on the conditions, delays in the transitions between states, stabilization of metastable states, or on the contrary enhancement of the choice of the most rewarding option induced by the time dependence of the parameter are identified.

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C⁰-normal form.

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant under ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry. The rich structure of these classical systems is investigated with geometric methods and the relation with the singularity theory approach is also highlighted. The geometric approach is the most straightforward way to obtain a general picture of the phase-space dynamics of the family as is defined by a complete subset in the space of control parameters complying with the symmetry constraint. It is shown how to find an energy-momentum map describing the phase-space structure of each member of the family, a catastrophe map that captures its global features and formal expressions for action-angle variables. Several examples, mainly taken from astrodynamics, are used as applications.

The article considers the question of local dynamics of the logistic equation with delays and the delay coefficients being the nonlinear functions. It is supposed that one of the parameters, characterizing the delay value, is sufficiently large. The stability criterion is formulated and critical cases in the problem of balance state stability are defined. The main content of the paper is focused on studying the local dynamics in cases close to critical. In nature, these critical cases are infinite dimensional. Special nonlinear boundary-value problems of the parabolic type are constructed, which play the role of normal forms. Their nonlocal dynamics defines the behavior of the solutions from the small neighborhood of the balance state of the initial equation. The conclusions on the role of nonlinearity in the delay coefficients, in the behavior of the dynamic properties of the solutions are summarized.