Bifurcations Of Periodic Orbits Research Articles

On Border-Collision Bifurcations in a Pulse System

Considering a piecewise smooth map describing the behavior of a pulse-modulated control system, we discuss border-collision related phenomena. We show that in the parameter space which corresponds to the domain of oscillatory mode a mapping is piecewise linear continuous. It is well known that in piecewise linear maps, classical bifurcations, for example, period doubling, tangent, fold bifurcations become degenerate (“degenerate bifurcations”), combining the properties of both smooth and border-collision bifurcations. We found unusual properties of this map, that consist in the fact that border-collision bifurcations of codimension one, including degenerate ones, occur when a pair of points of a periodic orbit simultaneously collides with two switching manifolds. This paper also discuss bifurcations of chaotic attractors such as merging and expansion (“interior”) crises, associated with homoclinic bifurcations of unstable periodic orbits.

Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability

Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle–node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.These dynamical insights enable us in turn to design a stochastic model whose solutions – as the delay parameter drifts slowly through its critical values – produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.

Bifurcations of periodic orbits in the generalised nonlinear Schrödinger Equation

Bifurcations of periodic orbits in the generalised nonlinear Schrödinger Equation

Bifurcation analysis of waning-boosting epidemiological models with repeat infections and varying immunity periods

We consider the SIRWJS epidemiological model that includes the waning and boosting of immunity via secondary infections. We carry out combined analytical and numerical investigations of the dynamics. The formulae describing the existence and stability of equilibria are derived. Combining this analysis with numerical continuation techniques, we construct global bifurcation diagrams with respect to several epidemiological parameters. The bifurcation analysis reveals a very rich structure of possible global dynamics. We show that backward bifurcation is possible at the critical value of the basic reproduction number, R0=1. Furthermore, we find stability switches and Hopf bifurcations from steady states forming multiple endemic bubbles, and saddle–node bifurcations of periodic orbits. Regions of bistability are also found, where either two stable steady states, or a stable steady state and a stable periodic orbit coexist. This work provides an insight to the rich and complicated infectious disease dynamics that can emerge from the waning and boosting of immunity.

Multiple switching and bifurcations of in-phase and anti-phase periodic orbits to chaotic coexistence in a delayed half-center CPG oscillator

Multiple switching and bifurcations of in-phase and anti-phase periodic orbits to chaotic coexistence in a delayed half-center CPG oscillator

Scenarios for the creation of hyperchaotic attractors in 3D maps

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map.

Saddle-node bifurcation of periodic orbit route to hidden attractors.

Hidden attractors are present in many nonlinear dynamical systems and are not associated with equilibria, making them difficult to locate. Recent studies have demonstrated methods of locating hidden attractors, but the route to these attractors is still not fully understood. In this Research Letter, we present the route to hidden attractors in systems with stable equilibrium points and in systems without any equilibrium points. We show that hidden attractors emerge as a result of the saddle-node bifurcation of stable and unstable periodic orbits. Real-time hardware experiments were performed to demonstrate the existence of hidden attractors in these systems. Despite the difficulties in identifying suitable initial conditions from the appropriate basin of attraction, we performed experiments to detect hidden attractors in nonlinear electronic circuits. Our results provide insights into the generation of hidden attractors in nonlinear dynamical systems.

Uniform bifurcation of comet-type periodic orbits in the restricted (n + 1)-body problem with non-Newtonian homogeneous potential

We treat the restricted (n+1)-body problem with a non-Newtonian homogeneous potential where the n primaries move on an arbitrary 2π-periodic orbit. We prove that the satellite equation has infinitely many periodic solutions that emerge from the infinity, asymptotically homothetic to the circular solutions of a central force problem. These solutions are obtained as critical solutions of a family of time-dependent perturbed Lagrangian systems, bifurcating uniformly from a compact set of periodic solutions of the unperturbed Lagrangian system.

Bifurcations of Grid-Following Rectifiers and Routes to Voltage Instability in Weak AC Grids

As indispensable components in modern power systems, grid-following rectifiers would interact with the grid. Such interaction, however, may lead to unexpected collapse of the ac bus voltage. In this paper, three routes to voltage instability are identified, namely, (i) supercritical Hopf bifurcation and successive saddle-node bifurcation of periodic orbits (SNPO); (ii) subcritical Hopf bifurcation; and (iii) departure from a stable basin of attraction. The SNPO and subcritical Hopf bifurcation are reported for the first time in rectifier-based power systems. It is also found that the rectifier-based system may undergo a Bautin bifurcation, which changes a Hopf bifurcation from supercritical to subcritical, or vice versa. The current loop of the grid-following rectifier has been found to play a crucial role in these bifurcation behaviors. Considering the inevitable and immediate voltage breakdown after the onset of a subcritical Hopf bifurcation, it is recommended to increase the bandwidth of the current loop to trigger the Bautin bifurcation, so that only supercritical Hopf bifurcation may occur. Furthermore, the original high-order and nonlinear model of the rectifier-based system can be simplified to a reduced order system by focusing on the center manifold. Based on the reduced system, the occurrence mode of Hopf bifurcation can be confirmed by inspecting the first Lyapunov coefficient. Finally, relevant bifurcation diagrams are derived, providing practical design guidelines to avoid an eventual system collapse.

Bifurcation and path-following continuation analysis of periodic orbits of an extended Fermi oscillator model

Bifurcation and path-following continuation analysis of periodic orbits of an extended Fermi oscillator model

Resonance, symmetry, and bifurcation of periodic orbits in perturbed Rayleigh–Bénard convection

This paper investigates the global structures of periodic orbits that appear in Rayleigh–Bénard convection, which is modelled by a two-dimensional perturbed Hamiltonian model, by focusing upon resonance, symmetry and bifurcation of the periodic orbits. First, we show the global structures of periodic orbits in the extended phase space by numerically detecting the associated periodic points on the Poincaré section. Then, we illustrate how resonant periodic orbits appear and specifically clarify that there exist some symmetric properties of such resonant periodic orbits which are projected on the phase space; namely, the period m and the winding number n become odd when an m-periodic orbit is symmetric with respect to the horizontal and vertical centre lines of a cell. Furthermore, the global structures of bifurcations of periodic orbits are depicted when the amplitude ɛ of the perturbation is varied, since in experiments the amplitude of the oscillation of the convection gradually increases when the Rayleigh number is raised.

On GIT quotients of the symplectic group, stability and bifurcations of periodic orbits (with a view towards practical applications)

On GIT quotients of the symplectic group, stability and bifurcations of periodic orbits (with a view towards practical applications)

Nonlinear model reduction for a cantilevered pipe conveying fluid: A system with asymmetric damping and stiffness matrices

We construct reduced-order models (ROMs) for a geometrically nonlinear viscoelastic cantilevered pipe conveying fluid, a dynamical system that includes asymmetric damping and stiffness matrices and nonlinear internal forces. The asymmetries of the damping and stiffness matrices are resulted from flow-induced gyroscopic and follower forces respectively. The ROMs are constructed using spectral submanifolds (SSMs). Specifically, we use a simulation-free approach implemented in SSMTool, an open-source package that automates the computation of SSMs, to construct the SSM-based ROMs. This approach takes equations of motion of the pipe as inputs and no simulation of the full system is involved. The ROMs enable efficient and accurate predictions of nonlinear dynamics of the system, including free vibration, isolas in forced response curves, bifurcations of periodic and quasi-periodic orbits, and heteroclinic and homoclinic orbits.

Hopf bifurcation at infinity in 3D Relay systems

A complete analysis of the limit cycle bifurcation from infinity in 3D Relay systems, which belong to the class of three-dimensional symmetric discontinuous piecewise linear systems with two zones, is presented. A criticality parameter is found, whose sign determines the character of the bifurcation. When such non-degeneracy parameter vanishes, a higher co-dimension bifurcation takes place, giving rise to the emergence of a curve of saddle–node bifurcations of periodic orbits, which allows to determine parameter regions where two limit cycles coexist.The existence of a large amplitude limit cycle that bifurcates from infinity is justified through a suitable adaptation of the closing equations method, and analytical expressions for its amplitude and period are provided. Derivatives of the corresponding transition maps are rigorously studied in order to characterize the stability of the bifurcating limit cycle.The theoretical results are applied to a specific family of 3D relay systems, where several high co-dimension bifurcation points are detected, organizing the bifurcation set of the family.

Invariant tori in dissipative hyperchaos.

One approach to understand the chaotic dynamics of nonlinear dissipative systems is the study of non-chaotic yet dynamically unstable invariant solutions embedded in the system's chaotic attractor. The significance of zero-dimensional unstable fixed points and one-dimensional unstable periodic orbits capturing time-periodic dynamics is widely accepted for high-dimensional chaotic systems, including fluid turbulence, while higher-dimensional invariant tori representing quasiperiodic dynamics have rarely been considered. We demonstrate that unstable 2-tori are generically embedded in the hyperchaotic attractor of a dissipative system of ordinary differential equations; tori can be numerically identified via bifurcations of unstable periodic orbits and their parameteric continuation and characterization of stability properties are feasible. As higher-dimensional tori are expected to be structurally unstable, 2-tori together with periodic orbits and equilibria form a complete set of relevant invariant solutions on which to base a dynamical description of chaos.

Bifurcation and Path-Following Analysis of Periodic Orbits of a Fermi Oscillator Model

In this research, we offer a bifurcation analysis to describe impacting, stick, and grazing between a particle and the piston to better understand the nonlinear dynamics of a Fermi oscillator. The principles of hybrid dynamical systems will be utilized to explain the moving process in such a Fermi oscillator. The motion complexity and stick mechanism of such an oscillator are demonstrated using periodic and chaotic motions. We introduce the hybrid dynamical system and numerical continuation detection tool COCO. Physical and mathematical models are used to construct the obtained bounded mathematical model. The frequency, amplitude in oscillating base displacement, and the gap between the stationary boundary and the piston’s equilibrium position are the major parameters. We employ path-following analysis to illustrate the bifurcation that leads to solution destabilization. From the viewpoint of eigenvalue analysis, the essence of period-doubling and Fold bifurcation is revealed. Numerical continuation methods are used to perform a complete one-parameter bifurcation analysis of the Fermi oscillator. The presence of codimension-1 bifurcations of limit cycles, like period-doubling, fold, and grazing bifurcations, is shown in this work. Then, the one-parameter continuation analysis is tracked in two-parameter bifurcation diagrams that include a second important factor of interest. Bifurcations cause all solutions to lose stability, according to our findings. The acquired results provide a better understanding of the Fermi oscillator mechanism and demonstrate that we may control the system dynamics by modifying the parameters.

Bifurcation of periodic orbits and its application for high-dimensional piecewise smooth near integrable systems with two switching manifolds

In this paper, we study the bifurcation of periodic orbits for high-dimensional piecewise smooth near integrable systems defined in three regions separated by two switching manifolds. We assume that the unperturbed system has a family of periodic orbits which cross two switching manifolds transversely. The expression of Melnikov function is derived based on the first integral. And the conditions of periodic orbits bifurcated from a family of periodic orbits for the high-dimensional piecewise smooth near integrable system are obtained. The theoretical results are applied to the bifurcation analysis of periodic orbits of two-degree-of-freedom piecewise smooth system of nonlinear energy sink. The periodic orbits configurations are presented with numerical method and the number of periodic orbits is three.

Two regularizations of the grazing-sliding bifurcation giving non equivalent dynamics

We present two ways of regularizing a one parameter family of piece-wise smooth dynamical systems undergoing a codimension one grazing-sliding global bifurcation of periodic orbits. First we use the Sotomayor-Teixeira regularization and prove that the regularized family has a saddle-node bifurcation of periodic orbits. Then we perform a hysteretic regularization and show that the regularized family has chaotic dynamics. Our result shows that, in spite that the two regularizations will give the same dynamics in the sliding modes, when a tangency appears the hysteretic process generates chaotic dynamics.

Bifurcation of Dividing Surfaces Constructed from Period-Doubling Bifurcations of Periodic Orbits in a Caldera Potential Energy Surface

In this work we analyze the bifurcation of dividing surfaces that occurs as a result of two period-doubling bifurcations in a 2D caldera-type potential. We study the structure, the range, the minimum and maximum extents of the periodic orbit dividing surfaces before and after a subcritical period-doubling bifurcation of the family of the central minimum of the potential energy surface. Furthermore, we repeat the same study for the case of a supercritical period-doubling bifurcation of the family of the central minimum of the potential energy surface. We will discuss and compare the results for the two cases of bifurcations of dividing surfaces.

The criticality of reversible quadratic centers at the outer boundary of its period annulus

This paper deals with the period function of the reversible quadratic centersXν=−y(1−x)∂x+(x+Dx2+Fy2)∂y, where ν=(D,F)∈R2. Compactifying the vector field to S2, the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle Πν at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of Xν that tend to Πν0 in the Hausdorff sense as ν→ν0. This notion is akin to the cyclicity in Hilbert's 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all ν=(D,F)∈R2 outside the segments {−1}×[0,1] and {0}×[0,2]. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all ν∈R2. In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert's 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result.