Degenerate Bifurcation Research Articles

Degenerate grazing bifurcations in a three-degree-of-freedom impact oscillator

This paper presents the analysis of the degenerate grazing bifurcation in a three-degree-of-freedom impact oscillator by studying the bifurcations of near-grazing period-one impact motion near the degenerate grazing point. Actually, this paper extends the higher-order zero time discontinuity mapping to perform the perturbation analysis of characteristic equation of period-one impact motion and obtains feasible eigenvalue approximation to study the potential bifurcations. The shooting method is applied to verify the validity of the derived approximation and corresponding computation results. In addition to the known bifurcation scenarios of saddle-node and period-doubling, novel Neimark–Sacker bifurcation and related co-dimension two bifurcation points of near-grazing period-one impact motion are also found to arise near the degenerate grazing point in a three-degree-of-freedom impact oscillator. For the in-depth understanding of near-grazing dynamics, the obtained results are compared with the reported results in the single- and two-degree-of-freedom impact oscillators.

Turing-Hopf bifurcation analysis in a superdiffusive predator-prey model.

The predator-prey model with superdiffusion is investigated in this paper. Here, the existence of Turing-Hopf bifurcation and the resulting dynamics are studied. To understand such a degenerate bifurcation in the anomalously diffusive system, the weakly nonlinear analysis is employed and the amplitude equations at the Turing-Hopf bifurcation point are obtained. Moreover, by analyzing the amplitude equations under suitable conditions, the abundant spatiotemporal dynamics are presented. In addition, to illustrate the theoretical analysis, some numerical simulations are carried out.

New dynamics coined in a 4-D quadratic autonomous hyper-chaotic system

This note revisits a 4-D quadratic autonomous hyper-chaotic system in Zarei and Tavakoli (2016) and mainly considers some of its rich dynamics not yet investigated: global boundedness, invariant algebraic surface, singularly degenerate heteroclinic cycle and limit cycle. The main contributions of the work are summarized as follows: Firstly, we prove that for 4a ≥ c > 2a > 0, d > 0 and e > 0 the solutions of that system are globally bounded by constructing a suitable Lyapunov function. Secondly, Q=x3−12ax12=0 is found to be one of invariant algebraic surfaces with the cofactor -4a for the model. Thirdly, numerical simulations for c=0 not only illustrate different types of infinitely many singularly degenerate heteroclinic cycles near which chaotic attractors or limit cycles generate, but also that some of more degenerate (in term of a pure imaginary pair, one zero and one negative eigenvalue) or stable (in sense of three negative eigenvalues and one null eigenvalue) non-isolated equilibria (0,0,x3,0)(x3∈R) directly change into the limit cycles or chaotic attractors with a small perturbation of c > 0, which is in the absence of singularly degenerate heteroclinic cycles and degenerate pitchfork bifurcation at the non-isolated equilibria. In particular, some kind of forming mechanism of Lorenz attractor and the hyper-chaotic attractor of that system with (a,b,c,d,e)=(10,28,83,1,16) is revealed, which are collapses of singularly degenerate heteroclinic cycles and explosions of stable non-isolated equilibria. Finally, circuit experiment implements the aforementioned hyper-chaotic attractor, showing very good agreement with the simulation results.

Suppression of grazing-induced instability in single degree-of-freedom impact oscillators

As a typical non-smooth bifurcation, grazing bifurcation can induce instability of elementary near-grazing impact periodic motion in impact oscillators. In this paper, the stability for near-grazing period-one impact motion to suppress grazing-induced instabilities is analyzed, based on which, a control strategy is proposed. The commonly-used leading order zero time discontinuity mapping is extended to a higher order one to aid the perturbation analysis of the characteristic equation. It is shown that the degenerate grazing bifurcation can eliminate the singular term in the characteristic equation, leading to bounded eigenvalues. Based on such a precondition, the bounded eigenvalues are further restricted inside the unit circle, and a continuous transition between non-impact and controlled impact motion is observed. One discrete feedback controller that changes the velocity of the oscillator based on the selected Poincare sections is adopted to demonstrate the control procedure.

Higher order zero time discontinuity mapping for analysis of degenerate grazing bifurcations of impacting oscillators

The discontinuity mapping technique provides quite a good way to study the near-grazing dynamics in the impacting oscillators. However, disagreements between the results obtained by the discontinuity mapping technique and the direct numerical observations still exist in terms of modeling and stability analysis. In particular, when applying the existing theory to deal with the complicated degenerate or other co-dimension two grazing bifurcations, even potential mistake in the qualitative analysis might arise. To reduce such disagreements, the local zero time discontinuity mapping is extended to higher order terms for the generic multi-degree-of-freedom impacting oscillators under harmonic excitation in this paper. Furthermore, to demonstrate the necessity and feasibility of using the higher order zero time discontinuity mapping for the analysis of degenerate grazing bifurcation, comparisons between the lower and higher order truncation are made through the existence analysis of near-grazing impact periodic motions.

Global Analysis of a Liénard System with Quadratic Damping

In this paper, the global analysis of a Liénard equation with quadratic damping is studied. There are 22 different global phase portraits in the Poincaré disc. Every global phase portrait is given as well as the complete global bifurcation diagram. Firstly, the equilibria at finite and infinite of the Liénard system are discussed. The properties of the equilibria are studied. Then, the sufficient and necessary conditions of the system with closed orbits are obtained. The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.

Two distinct excitable responses for a laser with a saturable absorber.

Excitable lasers with saturable absorbers are currently investigated as potential candidates for low level spike processing tasks in integrated optical platforms. Following a small perturbation of a stable equilibrium, a single and intense laser pulse can be generated before returning to rest. Motivated by recent experiments [Selmi et al., Phys. Rev. E 94, 042219 (2016)10.1103/PhysRevE.94.042219], we consider the rate equations for a laser containing a saturable absorber (LSA) and analyze the effects of different initial perturbations. With its three steady states and following Hodgkin classification, the LSA is a Type I excitable system. By contrast to perturbations on the intensity leading to the same intensity pulse, perturbations on the gain generate pulses of different amplitudes. We explain these distinct behaviors by analyzing the slow-fast dynamics of the laser in each case. We first consider a two-variable LSA model for which the conditions of excitability can be explored in the phase plane in a transparent manner. We then concentrate on the full three variable LSA equations and analyze its solutions near a degenerate steady bifurcation point. This analysis generalizes previous results [Dubbeldam et al., Phys. Rev. E 60, 6580 (1999)1063-651X10.1103/PhysRevE.60.6580] for unequal carrier density rates. Last, we discuss a fundamental difference between neuron and laser models.

Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems**This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11171185, 11571201).

This paper includes two parts. In the first part, we first focus on quasi-periodic time dependent perturbations of one-dimensional quasi-periodically forced systems with degenerate equilibrium. We study the system in two cases, for one of which system admits a response solution under a non-resonant condition on the frequency vector weaker than Brjuno–Rüssmann’s and for another of which system also admits a response solution without any non-resonant conditions. Next, we investigate the existence of response solutions of a quasi-periodic perturbed system with degenerate (including completely degenerate) equilibrium under Brjuno–Rüssmann’s non-resonant condition by using the Herman method. In the second part, we consider, firstly, the quasi-periodic perturbation of a universal unfolding of one-dimensional degenerate vector field . Secondly, we consider the perturbation of a universal unfolding of normal two-dimensional Hamiltonian system with completely degenerate equilibrium. With KAM theory and singularity theory, we show that these two classes of universal unfolding can persist on large Cantor sets under Brjuno–Rüssmann’s non-resonant condition, which implies all the invariant tori in the integrable part and all the bifurcation scenario can survive on large Cantor sets. The result for Hamiltonian system can apply directly to the response context for quasi-periodically forced systems. Our results in this paper can be regarded as an improvement with respect to several results in various literature (Broer et al 2005 Nonlinearity 18 1735–69; Broer et al 2006 J. Differ. Equ. 222 233–62; Wagener 2005 J. Differ. Equ. 216 216–81; Xu 2010 J. Differ. Equ. 250 551–71; Xu and Jiang 2010 Ergod. Theor. Dynam. Syst. 31 599–611; Lu and Xu 2014 Nonlinear Differ. Equ. Appl. 21 361–70).

Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk

We study a nonlinear Schrodinger equation with damping, detuning, and spatially homogeneous input terms, which is called the Lugiato-Lefever equation, on the unit disk with the Neumann boundary conditions. We aim at understanding bifurcations of a so-called cavity soliton which is a radially symmetric stationary spot solution. It is known by numerical simulations that a cavity soliton bifurcates from a spatially homogeneous steady state. We prove the existence of the parameter-dependent center manifold and a branch of radially symmetric steady state in a neighborhood of the bifurcation point. In order to capture further bifurcations of the radially symmetric steady state, we study a degenerate bifurcation for which two radially symmetric modes become unstable simultaneously, which is called the two-mode interaction. We derive a vector field on the center manifold in a neighborhood of such a degenerate bifurcation and present numerical simulations to demonstrate the Hopf and homoclinic bifurcations of bifurcating solutions.

Feedback Control of Grazing-Induced Chaos in the Single-Degree-of-Freedom Impact Oscillator

Based on the special dynamical property of continuous transition at certain degenerate grazing points in the single-degree-of-freedom impact oscillator, the control problem of the grazing-induced chaos is investigated in this paper. To design degenerate grazing bifurcations, we show how to obtain the degenerate grazing points of the 1/n impact periodic motions by the existence and stability analysis first. Then, a discrete-in-time feedback control strategy is used to suppress the grazing-induced chaos into the 1/n impact periodic motions precisely by the desired degenerate grazing bifurcation. The feasibility of the control strategy is verified by numerical simulations.

Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems

This paper deals with the problem of periodic orbit bifurcations for high-dimensional piecewise smooth systems. Under the assumption that the unperturbed system has a family of periodic orbits which are transversal to the switch plane, a formula for the first order Melnikov vector function is developed which can be used to study the number of periodic orbits bifurcated from the periodic orbits. We especially can use the function to study the number of periodic orbits both in degenerate Hopf bifurcations and in degenerate homoclinic bifurcations. Finally, we present two examples to illustrate an application of the theoretical results.

Effects of different dispersal patterns on the presence-absence of multiple species

Predicting which species will be present (or absent) across a geographical region remains one of the key problems in ecology. Numerous studies have suggested several ecological factors that can determine species presence-absence: environmental factors (i.e. abiotic environments), interactions among species (i.e. biotic interactions) and dispersal process. While various ecological factors have been considered, less attention has been given to the problem of understanding how different dispersal patterns, in interaction with other factors, shape community assembly in the presence of priority effects (i.e. where relative initial abundances determine the long-term presence-absence of each species). By employing both local and non-local dispersal models, we investigate the consequences of different dispersal patterns on the occurrence of priority effects and coexistence in multi-species communities. In the case of non-local, but short-range dispersal, we observe agreement with the predictions of local models for weak and medium dispersal strength, but disagreement for relatively strong dispersal levels. Our analysis shows the existence of a threshold value in dispersal strength (i.e. saddle-node bifurcation) above which priority effects disappear. These results also reveal a co-dimension 2 point, corresponding to a degenerate transcritical bifurcation: at this point, the transcritical bifurcation changes from subcritical to supercritical with corresponding creation of a saddle-node bifurcation curve. We observe further contrasting effects of non-local dispersal as dispersal distance changes: while very long-range dispersal can lead to species extinctions, intermediate-range dispersal can permit more outcomes with multi-species coexistence than short-range dispersal (or purely local dispersal). Overall, our results show that priority effects are more pronounced in the non-local dispersal models than in the local dispersal models. Taken together, our findings highlight the profound delicacy in the mediation of priority effects by dispersal processes: “big steps” can have more influence than many “small steps”.

Analytical determination for degenerate grazing bifurcation points in the single-degree-of-freedom impact oscillator

Co-dimension-one grazing bifurcations of 1 / n impact periodic motions are often accompanied by saddle-node bifurcations or period-doubling bifurcations. The presence of certain degenerate grazing bifurcation points plays an important role in transitions between these two different co-dimension-one grazing bifurcation scenarios. When the saddle-node bifurcation line and period-doubling bifurcation line meet at a grazing bifurcation point, a degenerate grazing bifurcation occurs. By considering the existence and stability of the 1 / n impact periodic motions near grazing points in single-degree-of-freedom impact oscillators, an analytical method is developed with the aid of the discontinuity mapping technique to determine the certain degenerate grazing bifurcation points of 1 / n motions. It is found that the linear term in the local discontinuity mapping does not affect the distribution of certain degenerate grazing points. The unfolding in the neighborhood of degenerate grazing bifurcation points is verified by numerical simulations.

Computation of the Unique Normal Form for Nonlinear Dynamical Systems in the Mechanical Applications

In the mechanical applications, research for degenerate bifurcations is usually connected with the normal form theory of nonlinear dynamical systems. In this paper, we mainly concern with the unique normal form for a class of three dimensional vector fields by the method of transformation with parameters. With the aid of the Maple software, a recursive formula for computing the coefficients of the normal form is given. Our results can be easily applied to investigate the direct bifurcation and stability of any model which can be written in this.

Saddle-node bifurcations of multiple homoclinic solutions in ODES

We study codimension 3 degenerate homoclinic bifurcations under periodic perturbations. Assume that among the 3 bifurcation equations, one is due to the homoclinic tangecy along the orbital direction. To the lowest order, the bifurcation equations become 3 quadratic equations. Under generic conditions on perturbations of the normal and tangential directions of the homoclinic orbit, up to 8 homoclinic orbits can be created through saddle-node bifurcations. Our results generate the homoclinic tangency bifurcation in Guckenheimer and Holmes [ 8 ].

Distribution of SS and AS and their bifurcations in aggregations of tuna around two FOBs

A number of empirical and theoretical studies shows that the exploitation of fish sources has benefitted a lot from artificial floating objects (abbr. FOBs) on the surface of ocean. In this paper we investigate the dynamical distribution in aggregations of tuna around two FOBs. We abandon the effort of precise computation for steady states and eigenvalues but utilize the monotonic intervals to determine the location of zeros and signs of eigenvalues qualitatively and use the symmetry of AS steady states to simplify the system. Our method enables us to find two more steady states than known results and complete the analysis of all steady states effectively. Furthermore, we display all bifurcations at steady states, including six bifurcations of co-dimension 1 and two bifurcations of co-dimension 2. One of bifurcations is a degenerate pitchfork bifurcation of co-dimension 4 but only a part of co-dimension 2 can be unfolded within the system. We construct sectorial regions to prove the nonexistence of closed orbits. Those results provide long-time prediction of steady numbers of tuna around the two FOBs and critical conditions for transitions of cases.

Numerical Investigation of the Hopf-Bogdanov-Takens Mode Interaction for a Fluid-conveying Tube

We investigate the dynamics after loss of stability of the downhanging configuration of a fluid conveying tube with a small end mass and an elastic support. By varying the fluid flow rate and the stiffness and location of the elastic support, different degenerate bifurcation scenarios can be observed. In this talk we investigate the bifurcating solution branches of the codimension 3 interaction between a Hopf bifurcation and a Takens-Bogdanov bifurcation. An elaboratory discussion of the primary and secondary solution branches was already given by Langford and Zhan. After reducing the system to the three-dimensional Normal Form equations we apply a numerical continuation procedure to locate the expected higher order bifurcation branches and possibly detect more complicated dynamics, like Shilnikov orbits.

Hopf‐Takens‐Bogdanov interaction for a fluid‐conveying tube

AbstractWe investigate the dynamics after loss of stability of the downhanging configuration of a fluid conveying tube with a small end mass and an elastic support.By varying the fluid flow rate and the stiffness and location of the elastic support, different degenerate bifurcation scenarios can be observed. In this article we investigate the bifurcating solution branches of the codimension 3 interaction between a Hopf bifurcation and a Bogdanov‐Takens bifurcation.A complete discussion of the primary and secondary solution branches was already given by W. F. Langford and K. Zhan. After reducing the system to the three‐dimensional Normal Form equations we apply a numerical continuation procedure to locate the expected higher order bifurcation branches and detect more complicated dynamics, like Shilnikov orbits. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Degenerate bifurcation of the rotating patches

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher m-fold symmetry with m≥3 the bifurcation does not occur. This answers to a problem left open in [10]. Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in [10]. The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation.

Triple-well potential with a uniform depth: Advantageous aspects in designing a multi-stable energy harvester

Analytical expressions for the bi- and tri-stable conditions of a multi-stable energy harvester (MEH) are derived on the basis of bifurcation analyses, and the associated multi-stable regions are characterized in a 2-D parametric space. It is found that a special boundary condition exists for a triple-well with a uniform depth (TU boundary condition), originating from a degenerate pitchfork bifurcation (DPF) point. Interestingly, the outermost well-to-well distance of the triple-well potential, when subjected to the condition that the maximum well depth is kept constant, becomes widest when the well depth is uniform. Accordingly, instead of investigating all possible parametric conditions, the design parameters for the optimal well configuration of the MEH can be sought most efficiently by simply tracing them on the TU boundary. A detailed examination of the potential well configurations along the TU boundary reveals that the most efficient energy harvesting from low-intensity ambient vibrations can be achieved on a TU boundary point, near the DPF point but inevitably a certain distance apart, by inducing an enlarged interwell motion. This investigation is experimentally validated.