Normal Form Coefficients Research Articles

Convergence analysis of a numerical method to solve the adjoint linearized periodic orbit equations

The equation adjoint to the linearization of the periodic orbit equations in a dynamical system is fundamental in the study of sensitivity issues for periodic orbits, e.g. in the synchronization of networks of weakly coupled oscillators. It is also fundamental in the computation of normal form coefficients for bifurcations of limit cycles. Numerically, the adjoint equations can be solved in a variety of ways. In the case where the periodic orbit equations are solved as a boundary value problem by collocation at Gauss points, a recent method allows one to compute the solution to the adjoint equations as a byproduct of Newton's method applied to solve the boundary value problem. This method is practically cost-free since it requires only the solution to an already factorized linear system. Moreover, it provides the solution to the adjoint equations in exactly the form needed in the applications. So far, the method has not been analyzed carefully and no rigorous convergence results have been proved. We prove that the method is equivalent to a collocation method for the adjoint equations so that convergence of order h m + 1 holds at all points and of order h 2 m at the points of the coarse mesh; here h is the maximum length of the mesh intervals and m is the degree of the approximating piecewise polynomials. We support this by extensive numerical tests.

PATTERN FORMATION IN A RING NETWORK WITH DELAY

We consider a ring network of three identical neurons with delayed feedback. Regarding the coupling coefficients as bifurcation parameters, we obtain codimension one bifurcation (including a Fold bifurcation and Hopf bifurcation) and codimension two bifurcations (including Fold–Fold bifurcations, Fold–Hopf bifurcations and Hopf–Hopf bifurcations). We also give concrete formulas for the normal form coefficients derived via the center manifold reduction that provide detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, and quasi-periodic solutions.

On stability at the Hamiltonian Hopf Bifurcation

For a 2 d.o.f. Hamiltonian system we prove the Lyapunov stability of its equilibrium with two double pure imaginary eigenvalues and non-semisimple Jordan form for the linearization matrix, when some coefficient in the 4th order normal form is positive (the equilibrium is known to be unstable, if this coefficient is negative). Such the degenerate equilibrium is met generically in one-parameter unfoldings, the related bifurcation is called to be the Hamiltonian Hopf Bifurcation. Though the stability is known since 1977, proofs that were published are either incorrect or not complete. Our proof is based on the KAM theory and a work with the Weierstrass elliptic functions, estimates of power series and scaling.

Calculation of coefficients of simplest normal forms of Neimark-Sacker and generalized Neimark-Sacker bifurcations

The coefficients of the simplest normal forms of both high-dimensional Neimark-Sacker and generalized Neimark-Sacker bifurcation systems were discussed. On the basis of the simplest normal form theory, using appropriate nonlinear transformations, direct computation and the second complete induction, the central manifold equations were further reduced to the simplest normal forms which only contained two nonlinear terms. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of the conventional normal forms of Neimark-Sacker and generalized Neimark-Sacker bifurcations systems.

Singular Hopf Bifurcation in Systems with Two Slow Variables

Hopf bifurcations have been studied intensively in two dimensional vector fields with one slow and one fast variable [É. Benoît et al., Collect. Math., 31 (1981), pp. 37–119; F. Dumortier and R. Roussarie, Mem. Amer. Math. Soc., 121 (577) (1996); W. Eckhaus, in Asymptotic Analysis II, Lecture Notes in Math. 985, Springer-Verlag, Berlin, 1983, pp. 449–494; M. Krupa and P. Szmolyan, SIAM J. Math. Anal., 33 (2001), pp. 286–314; J. Guckenheimer, in Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II Math. Phys. Chem. 137, Kluwer, Dordrecht, The Netherlands, 2004, pp. 295–316]. Canard explosions are associated with these singular Hopf bifurcations [S. M. Baer and T. Erneux, SIAM J. Appl. Math., 46 (1986), pp. 721–739; S. M. Baer and T. Erneux, SIAM J. Appl. Math., 52 (1992), pp. 1651–1664; B. Braaksma, J. Nonlinear Sci., 8 (1998), pp. 457–490; Y. Lijun and Z. Xianwu, J. Differential Equations, 206 (2004), pp. 30–54], manifested by a very rapid growth in the amplitude of periodic orbits. There has been less analysis of Hopf bifurcations in slow-fast systems with two slow variables where singular Hopf bifurcation occurs simultaneously with type II folded saddle-nodes [A. Milik and P. Szmolyan, in Multiple-Time-Scale Dynamical Systems, IMA Vol. Math. Appl. 122, Springer-Verlag, New York, 2001, pp. 117–140; M. Wechselberger, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 101–139]. This work contributes to our understanding of these Hopf bifurcations in five ways: (1) it computes the first Lyapunov coefficient of the bifurcation in terms of a normal form, (2) it describes global features of the flow that constrain the types of trajectories found in the system near the bifurcation, (3) it identifies codimension two bifurcations that occur as coefficients in the normal form vary, (4) it exhibits complex solutions that occur in the vicinity of the bifurcation for some values of the normal form coefficients, and (5) it identifies singular Hopf bifurcation as a mechanism for the creation of mixed-mode oscillations. A subtle aspect of the normal form is that terms of higher order contribute to the first Lyapunov coefficient of the bifurcation in an essential way.

Symbolic computation of the Birkhoff normal form in the problem of stability of the triangular libration points

The problem of stability of the triangular libration points in the planar circular restricted three-body problem is considered. A software package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is designed so that normalization problems of high analytical complexity could be solved. It is used to obtain the Birkhoff normal form of the Hamiltonian in the given problem. The normalization is carried out up to the 6th order of expansion of the Hamiltonian in the coordinates and momenta. Analytical expressions for the coefficients of the normal form of the 6th order are derived. Though intermediary expressions occupy gigabytes of the computer memory, the obtained coefficients of the normal form are compact enough for presentation in typographic format. The analogue of the Deprit formula for the stability criterion is derived in the 6th order of normalization. The obtained floating-point numerical values for the normal form coefficients and the stability criterion confirm the results by Markeev (1969) and Coppola and Rand (1989), while the obtained analytical and exact numeric expressions confirm the results by Meyer and Schmidt (1986) and Schmidt (1989). The given computational problem is solved without constructing a specialized algebraic processor, i.e., the designed computer algebra package has a broad field of applicability.

Two-parameter bifurcations in a network of two neurons with multiple delays

We consider a network of two coupled neurons with delayed feedback. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension 1 bifurcations (including a fold bifurcation and a Hopf bifurcation) and codimension 2 bifurcations (including fold–Hopf bifurcations and Hopf–Hopf bifurcations). We also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions. We also show how to evaluate critical normal form coefficients from the original system of delay-differential equations without computing the corresponding center manifolds.

Stable cycles in a Cournot duopoly model of Kopel

We consider a discrete map proposed by M. Kopel that models a nonlinear Cournot duopoly consisting of a market structure between the two opposite cases of monopoly and competition. The stability of the fixed points of the discrete dynamical system is analyzed. Synchronization of two dynamics parameters of the Cournot duopoly is considered in the computation of stability boundaries formed by parts of codim-1 bifurcation curves. We discover more on the dynamics of the map by computing numerically the critical normal form coefficients of all codim-1 and codim-2 bifurcation points and computing the associated two-parameter codim-1 curves rooted in some codim-2 points. It enables us to compute the stability domains of the low-order iterates of the map. We concentrate in particular on the second, third and fourth iterates and their relation to the period doubling, 1:3 and 1:4 resonant Neimark–Sacker points.

Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps

We discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont, a MATLAB toolbox for continuation and bifurcation analysis of dynamical systems. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points (i.e., limit point, period-doubling, and Neimark–Sacker) and their continuation in two control parameters, as well as detection and location of all codimension 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed, both numerically with finite directional differences and using symbolic derivatives of the original map. Using a parameter-dependent center manifold reduction, explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period. These asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations. We provide two examples illustrating the developed techniques: a generalized Hénon map and a juvenile/adult competition model from mathematical biology.

Remarks on interacting Neimark–Sacker bifurcations

We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue − 1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher-order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation of vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Representative nonsymmetric perturbations of the normal forms are studied numerically. Our case studies show a detailed picture near various bifurcation curves, which is richer than known theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds. We apply them to an example from robotics where we are able to demonstrate the existence of a bubble-structure, which was only observed in perturbed normal forms before.

The flutter of an airfoil with cubic structural and aerodynamic non-linearities

The flutter of a two-dimensional airfoil in a supersonic flow field, with cubic structural and aerodynamic non-linearities, is investigated using an efficient algorithm of normal form, which combines the normal form theory and the center manifold theory together. First, the stability of the linearized system is analyzed in the neighborhood of an equilibrium point, which shows that the flutter instability is resulted by the Hopf bifurcation. Then the normal form of Hopf bifurcation is deduced by applying the symbolic procedure of the new normal form algorithm to the perturbation equations. Analyzing the obtained coefficients of normal form shows that for a given system, the Hopf bifurcation can change from super-critical type to sub-critical type, consequently the flutter instability changes from “benign” type to “catastrophic” type, as the flight Mach number increases. Numerical simulations verify the dependence of response on initial conditions. Finally, the effects of the structural and aerodynamic parameters on the character of flutter instability are analyzed.

PRACTICAL COMPUTATION OF NORMAL FORMS ON CENTER MANIFOLDS AT DEGENERATE BOGDANOV–TAKENS BIFURCATIONS

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.

Numerical Normal Forms for Codim 2 Bifurcations of Fixed Points with at Most Two Critical Eigenvalues

In this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues. These include all strong resonances, as well as the degenerate flip and Neimark--Sacker bifurcations. Applying our results to n-dimensional maps, one avoids the computation of the center manifold, but one can directly evaluate the critical normal form coefficients in the original basis. The formulas remain valid also for n=2 and allow one to avoid the transformation of the linear part of the map into Jordan form. The developed techniques are tested on two examples: (1) a three-dimensional map appearing in adaptive control; (2) a periodically forced epidemic model. We compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models. To compute the required derivatives of the Poincaré map for the epidemic model, the automatic differentiation package ADOL-C is used.

Numerical Periodic Normalization for Codim 1 Bifurcations of Limit Cycles

Explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous ODEs are derived. They include second-order coefficients for the fold (limit point) bifurcation, as well as third-order coefficients for the flip (period-doubling) and Neimark--Sacker (torus) bifurcations. The formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0,T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the right-hand sides near the cycle. The formulas allow us to distinguish between sub- and supercritical bifurcations, in agreement with earlier asymptotic expansions of the bifurcating solutions. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The actual implementation is described in detail. We include three numerical examples, in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients.

A SUFFICIENT CONDITION FOR THE UNIQUENESS OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF GENERALIZED HOPF SINGULARITIES

A sufficient condition for the uniqueness of the Nth order normal form is provided. A new grading function is proposed and used to prove the uniqueness of the first-order normal forms of generalized Hopf singularities. Recursive formulas for computation of coefficients of unique normal forms of generalized Hopf singularities are also presented.

Chaos and Period-Adding; Experimental and Numerical Verification of the Grazing Bifurcation

Experimental results are presented for a single-degree-of-freedom horizontally excited pendulum that is allowed to impact with a rigid stop at a fixed angle θ to the vertical. By inclining the apparatus, the pendulum is allowed to swing in an effectively reduced gravity, so that for each fixed θ less than a critical value, a forcing frequency is found such that a period-one limit cycle motion just grazes with the stop. Experimental measurements show the immediate onset of chaotic dynamics and a period-adding cascade for slightly higher frequencies. These results are compared with a numerical simulation and continuation of solutions to a mathematical model of the system, which shows the same qualitative effects. From the model, the theory of discontinuity mappings due to Nordmark is applied to derive the coefficients of the square-root normal form map of the grazing bifurcation for this system. The grazing periodic orbit and its linearisation are found using a numerical continuation method for hybrid systems. From this, the normal-form coefficients are computed, which in this case imply that a jump to chaos and period-adding cascade occurs. Excellent quantitative agreement is found between the model simulation and the map, even over wide parameter ranges. Qualitatively, both accurately predict the experimental results, and after a slight change in the effective damping value, a striking quantitative agreement is found too.

THE FOLD-FLIP BIFURCATION

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.

Resonant triad dynamics in weakly damped Faraday waves with two-frequency forcing

Many of the interesting patterns seen in recent multi-frequency Faraday experiments can be understood on the basis of three-wave interactions (resonant triads). In this paper we consider two-frequency forcing and focus on a resonant triad occurring near the bicritical point where two pattern-forming modes with distinct wavenumbers onset simultaneously. This triad has been observed directly (in the form of rhomboids) and has also been implicated in the formation of quasipatterns and superlattices. We show how the symmetries of the undamped unforced problem (time translation, time reversal, and Hamiltonian structure) can be used, when the damping is weak, to obtain general scaling laws and additional qualitative properties of the normal form coefficients governing the pattern selection process near onset; such features help to explain why this particular triad is seen only for certain “low” forcing ratios, and predict the existence of drifting solutions and heteroclinic cycles. We confirm the anticipated parameter dependence of the coefficients and investigate its dynamical consequences using coefficients derived numerically from a quasipotential formulation of the Faraday problem due to Zhang and Viñals.

Double Hopf Bifurcations in the Differentially Heated Rotating Annulus

We study a mathematical model of the differentially heated rotating fluid annulus experiment. In particular, we analyze the double Hopf bifurcations that occur along the transition between axisymmetric steady solutions and nonaxisymmetric rotating waves. The model uses the Navier--Stokes equations in the Boussinesq approximation. At the bifurcation points, center manifold reduction and normal form theory are used to deduce the local behavior of the full system of partial differential equations from a low-dimensional system of ordinary differential equations. It is not possible to compute the relevant eigenvalues and eigenfunctions analytically. Therefore, the linear part of the equations is discretized, and the eigenvalues and eigenfunctions are approximated from the resulting matrix eigenvalue problem. However, the projection onto the center manifold and reduction to normal form can be done analytically. Thus, a combination of analytical and numerical methods is used to obtain numerical approximations of the normal form coefficients, from which the dynamics are deduced. The results indicate that, close to the transition, there are regions in parameter space where there are multiple stable waves. Hysteresis of these waves is predicted. The validity of the results is shown by their consistency with experimental observations.

Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry

We consider a general class of first-order nonlinear delay-differential equations (DDEs) with reflectional symmetry, and study completely the bifurcations of the trivial equilibrium under some generic conditions on the Taylor coefficients of the DDE. Our analysis reveals a Hopf bifurcation curve terminating on a pitchfork bifurcation line at a codimension two Takens–Bogdanov point in parameter space. We compute the normal form coefficients of the reduced vector field on the centre manifold in terms of the Taylor coefficients of the original DDE, and in contrast to many previous bifurcation analyses of DDEs, we also compute the unfolding parameters in terms of these coefficients. For application purposes, this is important since one can now identify the possible asymptotic dynamics of the DDE near the bifurcation points by computing quantities which depend explicitly on the Taylor coefficients of the original DDE. We illustrate these results using simple model systems relevant to the areas of neural networks and atmospheric physics, and show that the results agree with numerical simulations.