Degenerate Bifurcation Research Articles

Stability of weakly-damped systems under follower loads exhibiting Hopf bifurcations

The existence of flutter instability through a Hopf (degenerate or generic) bifurcation prior to divergence (static) instability of Ziegler’s cantilever, weakly-damped models under partial follower compressive tip load (autonomous systems) is thoroughly discussed. Attention is focused on establishing the locus of the above dynamic bifurcations in regions of divergence defined by the nonconservative loading parameter η. The local as well as the global nonlinear dynamic instability of such bifurcations is conveniently established using the Runge–Kutta numerical scheme. Several examples for a variety of values of parameters will demonstrate the proposed technique as well as the new findings presented herein.

Imperfection sensitivity of degenerate hilltop branching points

Imperfection sensitivity is investigated for a degenerate hilltop branching point, where a degenerate bifurcation point exists at a limit point. A degenerate hilltop branching point is important as it is a byproduct of optimization of shallow shell structures under non-linear buckling constraints. A systematic procedure is presented for asymptotic sensitivity analysis based on enumeration of vertices of a convex region defined by linear inequality constraints on the orders of the variables. The effectiveness of the proposed method is demonstrated by sensitivity analysis of degenerate hilltop branching points, considering minor and major imperfections, corresponding to an unstable-symmetric or asymmetric bifurcation point at the limit point. It is found that a hilltop branching point can be imperfection sensitive.

Computation of bifurcation manifolds of linearly independent homoclinic orbits

Regarding the small perturbation as a parameter in an appropriate space of functions, we can discuss co-existence of homoclinic orbits for non-autonomous perturbations of an autonomous system in R n and describe conditions of parameters for such degenerate homoclinic bifurcations with some bifurcation manifolds of infinite dimension. Since those manifolds determine the relation among parameters for such bifurcations, in this paper we give an algorithm to compute approximately those manifolds and concretely obtain their first order approximates.

Pattern formation in forced reaction diffusion systems with nearly degenerate bifurcations

The existence and stability of stable standing-wave patterns in an assembly of spatially distributed generic oscillators governed by a couple of complex Ginzburg-Landau equations, subjected to parametric forcing, are reported. The mechanism of a dispersion-induced pattern in dissipative oscillators parametrically forced near the degenerate Turing-Hopf bifurcation is also illustrated. We show that, when excitation occurs just after the Turing bifurcation and before the Hopf instability, the system exhibits a new type of stable standing-wave structures, namely the mixed-mode solutions. The Brussellator-model, parametrically forced below the threshold of oscillations, is analyzed as an example of calculation.

Co-dimension-Two Grazing Bifurcations in Single-Degree-of-Freedom Impact Oscillators

Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.

Rosette Central Configurations, Degenerate Central Configurations and Bifurcations

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m 1 lie at the vertices of a regular n-gon, n particles of mass m 2 lie at the vertices of another n-gon concentric with the first, but rotated of an angle π /n, and an additional particle of mass m 0 lies at the center of mass of the system. This system admits two mass parameters μ = m 0/m 1 and e = m 2/m 1. We show that, as μ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every e > 0, while if n = 3 there is a bifurcation only for some values of e.

DEGENERATE BIFURCATION SCENARIOS IN THE DYNAMICS OF COUPLED OSCILLATORS WITH SYMMETRIC NONLINEARITIES

We study the degenerate bifurcations of nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. The potentials of both the oscillator and the coupling spring are adopted to be even-power polynomials with non-negative coefficients. Coupling parameter ε is defined and the bifurcations of the nonlinear normal modes structure with change of this coupling parameter are revealed. The degeneracy in the dynamics is manifested by a "bifurcation from infinity" where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Other (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1 : 1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by the mechanism of successive cusp catastrophes with the growth of coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincaré sections). In the particular case of pure cubic nonlinearity of the oscillator and the coupling spring, an agreement between quantitative analytical predictions and numerical results is observed.

Unfolding degenerate grazing dynamics in impact actuators

In this paper, a selected analysis of the dynamics in an example impact microactuator is performed through a combination of numerical simulations and local analysis. Here, emphasis is placed on investigating the system response in the vicinity of the so-called grazing trajectories, i.e. motions that include zero-relative-velocity contact of the actuator parts, using the concept of discontinuity mappings that account for the effects of low-relative-velocity impacts and brief episodes of stick–slip motion. The analysis highlights the existence of isolated co-dimension-two grazing bifurcation points and the way in which these organize the behaviour of the impacting dynamics. In particular, it is shown how higher-order truncations of local maps of the near-grazing dynamics predict and enable the computation of global bifurcation curves emanating from such degenerate bifurcation points, thereby unfolding the near-grazing dynamics. Although the numerical results presented here are specific for the chosen model of an electrically driven and previously experimentally realized impact microactuator, the methodology generalizes naturally to arbitrary systems with impacts. Moreover, the qualitative nature of the near-grazing dynamics is expected to generalize to systems with similar nonlinearities.

On the quasi-periodic [formula omitted]-fold degenerate bifurcation

This paper analyses the d -fold degenerate bifurcation of invariant quasi-periodic tori of normal dimension one. It is shown that for integrable families of diffeomorphisms unfolding a d -fold degenerate invariant torus versally, invariant Diophantine tori in the family persist under a small non-integrable perturbation. As a consequence, a subset of positive measure (with respect to an interpolating smooth manifold) of the bifurcation set persists as well. The proof is a consequence of the translated torus theorem of Rüssmann and Herman.

Local bifurcations and codimension-3 degenerate bifurcations of a quintic nonlinear beam under parametric excitation

This paper presents the analysis of the local and codimension-3 degenerate bifurcations in a simply supported flexible beam with quintic nonlinear terms subjected to a harmonic axial excitation for the first time. The quintic nonlinear equation of motion with parametric excitation is derived using the Hamilton’s principle. The parametrically excited system is transformed to the averaged equations using the method of multiple scales. Numerical method is used to compute the bifurcation response curves based on the averaged equations. The investigations are made on the effects of quintic nonlinear terms and parametric excitation on the local bifurcations. The stability of trivial solution is analyzed. With the aid of normal form theory, the explicit expressions are obtained for normal form associated with a double zero eigenvalues and Z2-symmetry of the averaged equations. Based on normal form, the analysis of codimension-3 degenerate bifurcations is performed for a simply supported quintic nonlinear beam with the focus on homoclinic and heteroclinic bifurcations. It is found from the analysis of homoclinic and heteroclinic bifurcations that multiple limit cycles may simultaneously exist for quintic nonlinearity. In particular, the number of limit cycles can be precisely determined analytically. New jumping phenomena are discovered in amplitude modulated oscillations.

Classification of compatibility paths of SDOF mechanisms

The compatibility paths of mechanisms with a single degree-of-freedom typically form sets of curves in the global representation space. We classify the different cases of compatibility by introducing an energy function. The result obtained also depends on which element of the mechanism is regarded as driven. The different singularity types are demonstrated by examples (split-vanish point, limit point, asymmetric bifurcation, infinitely degenerate bifurcation, hilltop point, compatibility surface).

Control of integrable Hamiltonian systems and degenerate bifurcations.

We discuss control of low-dimensional systems which, when uncontrolled, are integrable in the Hamiltonian sense. The controller targets an exact solution of the system in a region where the uncontrolled dynamics has invariant tori. Both dissipative and conservative controllers are considered. We show that the shear flow structure of the undriven system causes a Takens-Bogdanov bifurcation to occur when control is applied. This implies extreme noise sensitivity. We then consider an example of these results using the driven nonlinear Schrödinger equation.

Degenerate bifurcation points of periodic solutions of autonomous Hamiltonian systems

We study connected branches of nonconstant 2 π-periodic solutions of the Hamilton equation x ̇ (t)=λJ ∇H(x(t)), where λ∈(0,+∞), H∈C 2( R n× R n, R) and ∇ 2H(x 0)= A 0 0 B for x 0∈∇ H −1(0). The Hessian ∇ 2 H( x 0) can be singular. We formulate sufficient conditions for the existence of such branches bifurcating from given ( x 0, λ 0). As a consequence we prove theorems concerning the existence of connected branches of arbitrary periodic nonstationary trajectories of the Hamiltonian system x ̇ (t)=J ∇H(x(t)) emanating from x 0. We describe also minimal periods of trajectories near x 0.

Delay dynamical systems and applications to nonlinear machine-tool chatter

The stability behaviour of machine chatter that exhibits Hopf and degenerate bifurcations has been examined without the assumption of small delays between successive cuts. Delay dynamical system theory leading to the reduction of the infinite-dimensional character of the governing delay differential equations (DDEs) to a finite-dimensional set of ordinary differential equations have been employed. The essential mathematical arguments for these systems in the context of retarded DDEs are summarized. Then the application of these arguments in the stability study of machine-tool chatter with multiple time delays is presented. Explicit analytical expressions ensuring stable and unstable machining when perturbations are periodic, stochastic and nonlinear have been derived using the integral averaging method and Lyapunov exponents.

A Degenerate bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes(NNMs) of an unforced system consisting of a linear oscillator weaklycoupled to a nonlinear one that possesses essential stiffnessnonlinearity. By defining the small coupling parameter e, we study thedynamics of this system at the limit e → 0. The degeneracy in the dynamics ismanifested by a 'bifurcation from infinity' where a bifurcation point isgenerated at high energies, as perturbation of a state of infiniteenergy. Another (nondegenerate) bifurcation point is generated close tothe point of exact 1:1 internal resonance between the linear andnonlinear oscillators. The degenerate bifurcation structure can bedirectly attributed to the high degeneracy of the uncoupled system inthe limit e → 0, whose linearized structure possesses a double zero, and aconjugate pair of purely imaginary eigenvalues. First we construct localanalytical approximations to the NNMs in the neighborhoods of thebifurcation points and at other energy ranges of the system. Then, we`connect' the local approximations by global approximants, and identifyglobal branches of NNMs where unstable and stable mode and inverse modelocalization between the linear and nonlinear oscillators take place fordecreasing energy.

Stability conditions for the traveling pulse: Modifying the restitution hypothesis.

As a simple model of reentry, we use a general FitzHugh-Nagumo model on a ring (in the singular limit) to build an understanding of the scope of the restitution hypothesis. It has already been shown that for a traveling pulse solution with a phase wave back, the restitution hypothesis gives the correct stability condition. We generalize this analysis to include the possibility of a pulse with a triggered wave back. Calculating the linear stability condition for such a system, we find that the restitution hypothesis, which depends only on action potential duration restitution, can be extended to a more general condition that includes dependence on conduction velocity restitution as well as two other parameters. This extension amounts to unfolding the original bifurcation described in the phase wave back case which was originally understood to be a degenerate bifurcation. In addition, we demonstrate that dependence of stability on the slope of the restitution curve can be significantly modified by the sensitivity to other parameters (including conduction velocity restitution). We provide an example in which the traveling pulse is stable despite a steep restitution curve. (c) 2002 American Institute of Physics.

Clustering in globally coupled oscillators

In this paper we describe approaches to clustering in systems of globally coupled identical oscillators. The first of these approaches is based entirely on the symmetry of such systems, and gives information about the behaviour of the systems near degenerate bifurcation points. We summarize existing results from such analysis, and indicate why further techniques are required to augment the symmetry-based methods. This leads to a second approach based on constructing certain reduced models. This modelling approach relies indirectly on symmetry, using the fact that the systems in question have many invariant subspaces as a result of their symmetry. It is shown how knowledge of behaviour on certain subspaces can be used to predict behaviour on others subspaces, even when their dimensions are different. In applications, this approach can be used to predict the stable clustering behaviour that cannot be predicted by other approaches and may be hard to find numerically. All results are illustrated with examples.

Bifurcations in a Mathieu equation with cubic nonlinearities: Part II

In a previous paper [Chaos Solitons Fract. 14(2) (2002) 173], the authors investigated the dynamics of the equation: d 2x dt 2 +(δ+ϵ cost)x+ϵ Ax 3+Bx 2 dx dt +Cx dx dt 2+D dx dt 3 =0 We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the resulting slow flow and some of the bifurcations near this point were looked at. In this work we present additional results concerning the bifurcations around this point using analytic techniques and AUTO. An analytic approximation for a heteroclinic bifurcation curve is obtained. Additional results on the bifurcations of periodic orbits in the slow flow are also presented.

RESONANCE STRUCTURE IN A WEAKLY DETUNED LASER WITH INJECTED SIGNAL

We describe the qualitative dynamics and bifurcation set for a laser with injected signal for small cavity detunings. The main organizing center is the Hopf-saddle-node bifurcation from where a secondary Hopf bifurcation of a periodic orbit originates. We show that the laser's stable cw solution existing for low injections, also suffers a secondary Hopf bifurcation. The resonance structure of both tori interact, and homoclinic orbits to the "off" state are found inside each Arnold tongue. The accumulation of all the above resonances towards the Hopf-saddle-node singularity points to the occurrence of a highly degenerate global bifurcation at the codimension-2 point.

DYNAMICS OF AUTOPARAMETRIC VIBRATION ABSORBERS USING MULTIPLE PENDULUMS

This paper analyzes the dynamics of a resonantly excited single-degree-of-freedom linear system coupled to an array of non-linear autoparametric vibration absorbers (pendulums). The case of a 1:1:…:2 internal resonance between pendulums and the primary oscillator is studied. The method of averaging is used to obtain first order approximations to the non-linear response of the system. The stability and bifurcations of equilibria of the averaged equations are computed. It is shown that the frequency interval of the unstable single-mode response, or the absorber bandwidth, can be enlarged substantially compared to that of a single pendulum absorber by adjusting individually the internal mistunings of the pendulums. Use of multiple pendulums is also shown to engender degenerate bifurcations as the double-mode response “switches” from one pendulum to the other with changing external excitation frequency. The effect of various parameters on the performance is discussed and a strategy is developed to find the most effective parameters for maximum bandwidth of operation. This results in a significant enhancement of the performance of autoparametric vibration absorbers.