Takens Normal Form Research Articles

Degenerate Bogdanov\u2013Takens bifurcations in a bulk viscous cosmology

Using the dynamical system theory we show that the Friedmann–Robertson–Walker (FRW) cosmological model with bulk viscous fluid in the presence of cosmological constant is equivalent to a degenerate two dimensional Bogdanov–Takens normal form. The equation of state parameter, omega , the bulk viscosity coefficient, xi , and the cosmological constant, Lambda , define the necessary parameters for unfolding the degenerate Bogdanov-Takens system. The fixed points of the system are discussed together with the variation of their stability properties upon changing the relevant parameters omega , Lambda and xi . The variation of the stability properties are visualized by the appropriate bifurcation diagrams. Phase portrait for finite domain and global phase portrait are displayed and the issue of the structural stability is discussed. Typical issues such as late acceleration or inflation that can be induced by viscosity and could have relevance to observational cosmology are also discussed.

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n \geq 3$. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensure that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a nilpotent version of Bruno's condition (A). In dimension 2, no condition is required since, according to Str{\'o}zyna- Zoladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\frak{sl}\_2(\Bbb C)$-representations.

Center conditions for nilpotent cubic systems using the Cherkas method

In this study, we consider the center problem of a cubic polynomial differential system with a nilpotent linear part. The analysis is based on the application of the Cherkas method to the Takens normal form. The analysis requires many computations, which are verified by employing one algebraic manipulator and extensive use of the computer algebra system called Singular.

The Bogdanov–Takens Normal Form: A Minimal Model for Single Neuron Dynamics

Conductance-based (CB) models are a class of high dimensional dynamical systems derived from biophysical principles to describe in detail the electrical dynamics of single neurons. Despite the high dimensionality of these models, the dynamics observed for realistic parameter values is generically planar and can be minimally described by two equations. In this work, we derive the conditions to have a Bogdanov–Takens (BT) bifurcation in CB models, and we argue that it is plausible that these conditions are verified for experimentally-sensible values of the parameters. We show numerically that the cubic BT normal form, a two-variable dynamical system, exhibits all of the diversity of bifurcations generically observed in single neuron models. We show that the Morris–Lecar model is approximately equivalent to the cubic Bogdanov–Takens normal form for realistic values of parameters. Furthermore, we explicitly calculate the quadratic coefficient of the BT normal form for a generic CB model, obtaining that by constraining the theoretical I-V curve’s curvature to match experimental observations, the normal form appears to be naturally cubic. We propose the cubic BT normal form as a robust minimal model for single neuron dynamics that can be derived from biophysically-realistic CB models.

Accurate Approximation of Homoclinic Solutions in Gray–Scott Kinetic Model

The second-order predictor for the homoclinic orbit is applied to the Gray–Scott model. The problem is used to illustrate the approximation of the homoclinic orbits near a generic Bogdanov–Takens bifurcation in n-dimensional systems of differential equations. In the process, we show that it is necessary to take (usually ignored) cubic terms in the Bogdanov–Takens normal form into account to derive a correct second-order prediction for the homoclinic bifurcation curve. The analytic solutions are compared with those obtained by numerical continuation.

Divergence of the reduction to the multidimensional nilpotent Takens normal form

In Stróżyna and Żoładek (2008 Bull. Belgian Math. Soc. Simon Stevin 15 927–34) a generalization of the Takens normal form for a nilpotent singularity of a vector field in was obtained. Here we present an example where the corresponding normalizing series is divergent. This indicates that the generalized Takens normal form for a general nilpotent singularity in , n ⩾ 3, is non-analytic.

Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms**In memoriam: Floris Takens.

Dynamical phenomena are studied near a Hopf-saddle-node bifurcation of fixed points of 3D-diffeomorphisms. The interest lies in the neighbourhood of weak resonances of the complex conjugate eigenvalues. The 1 : 5 case is chosen here because it has the lowest order among the weak resonances, and therefore it is likely to have a most visible influence on the bifurcation diagram. A model map is obtained by a natural construction, through perturbation of the flow of a Poincaré–Takens normal form vector field. Global bifurcations arise in connection with a pair of saddle-focus fixed points: homoclinic tangencies occur near a sphere-like heteroclinic structure formed by the 2D stable and unstable manifolds of the saddle points. Strange attractors occur for nearby parameter values and three routes are described. One route involves a sequence of quasi-periodic period doublings of an invariant circle where loss of reducibility also takes place during the process. A second route involves intermittency due to a quasi-periodic saddle-node bifurcation of an invariant circle. Finally a route involving heteroclinic phenomena is discussed. Multistability occurs in several parameter subdomains: we analyse the structure of the basins for a case of coexistence of a strange and a quasi-periodic attractor and for coexistence of two strange attractors. By construction, the phenomenology of the model map is expected in generic families of 3D diffeomorphisms.

Unique normal forms for area preserving maps near a fixed point with neutral multipliers

We study normal forms for families of area-preserving maps which have a fixed point with neutral multipliers -1 or +1 at epsilon=0. Our study covers both the orientation-preserving and orientation-reversing cases. In these cases Birkhoff normal forms do not provide a substantial simplification of the system. In the paper we prove that the Takens normal form vector field can be substantially simplified. We also show that if certain non-degeneracy conditions are satisfied no further simplification is generically possible since the constructed normal forms are unique. In particular, we provide a full system of formal invariants with respect to formal coordinate changes.

Geometry and dynamics of mildly degenerate Hopf–Neĭmarck–Sacker families near resonance

This paper deals with families of planar diffeomorphisms undergoing a Hopf–Neĭmarck–Sacker bifurcation and focuses on bifurcation diagrams of the periodic dynamics. In our universal study the corresponding geometry is classified using Lyapunov–Schmidt reduction and contact-equivalence singularity theory, equivariant under an appropriate cyclic group. This approach recovers the non-degenerate standard Arnol'd resonance tongues. Our main concern is a mildly degenerate case, for which we analyse a 4-parameter universal model in detail. The corresponding resonance set has a Whitney stratification, which we explain by giving its incidence diagram and by giving both two- and three-dimensional cross-sections of parameter space. We investigate the further complexity of the dynamics of the universal model by performing a bifurcation study of an approximating family of Takens normal form vector fields. In particular, this study demonstrates how the bifurcation set extends an approximation of the resonance set of the universal model.

Unique resonant normal forms for area-preserving maps at an elliptic fixed point

We study possible simplifications of normal forms for area-preserving maps near a resonant elliptic fixed point. In the generic case we prove that at all orders the Takens normal form vector field can be transformed to the particularly simple form described by the formal interpolating Hamiltonian The form of formal series A and B depends on the order of the resonance n. For each n ⩾ 3 we establish which terms of the series can be eliminated by a canonical substitution and derive a unique normal form which provides a full set of formal invariants with respect to canonical changes in coordinates.We extend these results to families of area-preserving maps. Then the formal interpolating Hamiltonian takes a form similar to the case of an individual map but involves formal power series in action I and the parameter.

Multidimensional formal Takens normal form

We present a multidimensional analogue of the classical Takens normal form for a nilpotent singularity of a vector field.

The Analytic and Formal Normal Form for the Nilpotent Singularity

We study orbital normal forms for analytic planar vector fields with nilpotent singularity. We show that the Takens normal form is analytic. In the case of generalized cusp we present the complete formal orbital normal form; it contains functional moduli. We interprete the coefficients of these moduli in terms of the hidden holonomy group.

THE STABILITY OF SPATIALLY NONHOMOGENEOUS STEADY STATES IN A TUBULAR CHEMICAL REACTOR

We study the axial heat and mass transfer in a highly diffusive tubular chemical reactor in which a simple reaction is occurring. The steady state solutions of the governing equations are studied using matched asymptotic expansions, the theory of dynamical systems, and by calculating the solutions numerically. In particular, the effect of varying the Peclet and Damköhler numbers (P and D) is investigated. A simple expression for the approximate location of the transition layer for large Peclet number is derived and its accuracy tested against the numerical solution. The stability of the steady states is examined by calculating the eigenvalues and eigenfunctions of the linearized equations. It is shown that a Hopf bifurcation of the CSTR model (i.e., the limit as the P approaches zero) can be continued up to order 1 in the Peclet number. Furthermore, it is shown numerically that for appropriate values of the Peclet number, the Damköhler number, and B (the heat of reaction) these Hopf bifurcations merge with the limit points of an "S–shaped" bifurcation curve in a higher order singularity controlled by the Bogdanov–Takens normal form. Consequently, there must exist a finite amplitude, nonuniform, stable periodic solution for parameter values near this singularity. The existence of higher order degeneracies is also explored. In particular, it is shown for D ≪ 1 that no value of P exists where two pairs of complex conjugate eigenvalues of the steady state solutions can cross the imaginary axis simultaneously.

Normal forms for nonlinear vector fields. I. Theory and algorithm

Normal forms are analytical tools for studying the qualitative behavior of the nonlinear vector fields. A tutorial for the nonspecialist in general, and the circuit theorist in particular, on the basic concept and foundation of the modern theory of normal forms for nonlinear vector fields, is presented. After stating the Poincare and the Takens normal form, the latest refinements due to S. Ushiki (1984) are pointed out. For pedagogical reasons, the familiar Jordon form is first derived and shown to be appropriate normal form for matrices. Rather than using a standard linear approach, formulation is based on the 'method of infinitesimal deformation' which generalizes to nonlinear vector fields. >

A codimension three bifurcation for the laser with saturable absorber

We study the five mode equations which are used to model the dynamical behavior of a laser with saturable absorber in the mean field limit and exact resonance. We show that in this system a codimension three bifurcation exists where a tricritical point of the stationary solution encounters a double zero eigenvalue. A center manifold reduction is performed to fix the three-dimensional submanifold in parameter space where this degeneracy occurs. The associated Takens-normal form is given. By unfolding the normal form we obtain all structurally stable phase portraits near this bifurcation point and display them in the form of bifurcation diagrams with the laser pumping rate as a distinguished bifurcation parameter. These diagrams allow a unifying analytical and geometrical description of many different numerical solutions of the equations describing a laser with absorber. In particular, they yield the connection of the small amplitude periodic solutions with passiveQ-switching and suggest new bifurcation processes, which one can expect to occur for physical parameters near the critical submanifold. The existence of a codimension four bifurcation is indicated.