Bifurcations Of Periodic Orbits Research Articles

Bifurcation of periodic orbits in discontinuous systems

This paper treads discontinuous bifurcation in piecewise smooth systems of Filippov type. These bifurcations occur when a fixed point or a periodic orbit crosses with the border between two regions of smooth behavior. A detailed analysis of generalization Poincare map and monodromy matrix which are related shows that subfamily of system with invariant cone-like objects is foliated by periodic orbits and determines its stability. In addition, we introduce a theoretical framework for analyzing 3D perturbed nonlinear piecewise smooth systems and give necessary conditions so that different types of bifurcations occur. The analysis identifies criteria for the existence of a novel bifurcation based on sensitively the location of the return map. Moreover, the piecewise smooth Melnikov function and sufficient conditions of the existence of the periodic orbits for nonlinear perturbed system are explicitly obtained.

Complex non‐linear phenomena and stability analysis of interconnected power converters used in distributed power systems

Distributed power systems are considered to be a key element of future power grids. Their main characteristic is the local production and consumption of energy that is accomplished using a number of power converters that interconnect local sources and loads. In this paper, a current mode controlled dc-dc converter which feeds another current mode controlled dc-dc converter is considered to meet the demands of the load. It shows that the existence of the second converter at the output has destabilising effect on the overall system. Such a system may also exhibit interactions of various types of instability that conventional modeling methods cannot predict. Slow-timescale oscillation in standalone switching converters is usually attributed to the occurrence of a Neimark-Sacker bifurcation of the fundamental periodic orbit of the discrete-time model of the system. But in the studied cascaded system, the underlying mechanism is quite different. This paper fully explains the mechanisms as well as the conditions of the occurrence of such instabilities by employing the analytical and numerical tools of the exact switched model of the system. These results can be useful for developing design guidelines to avoid such problems. Finally, the results have been validated experimentally.

Multiple Bifurcations of a Cylindrical Dynamical System

Abstract This paper focuses on multiple bifurcations of a cylindrical dynamical system, which is evolved from a rotating pendulum with SD oscillator. The rotating pendulum system exhibits the coupling dynamics property of the bistable state and conventional pendulum with the ho- moclinic orbits of the first and second type. A double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are detected by using analytical analysis and nu- merical calculation. It is found that the homoclinic orbits of the second type can bifurcate into a pair of rotational limit cycles, coexisting with the oscillating limit cycle. Additionally, the results obtained herein, are helpful to explore different types of limit cycles and the complex dynamic bifurcation of cylindrical dynamical system.

Backbones in the parameter plane of the Hénon map.

Parameter plane (b, a) of the real Hénon map has been investigated for curves of bifurcation, curves of homoclinic heteroclinic onsets, and also searching for borders of areas variously characterized. Such curves are, in general, complicated and show singularities. Pieces of two monotone curves, spanning the (b, a) parameter plane of the real Hénon map, can be detected in four quite different studies appeared along the years 1982-2008. We study the extent of their similarity to read and interpret them into the same curves. To us, these two curves are the accumulation loci of bifurcation curves of two principal families of periodic sinks of type "period-adding machine." We call them "backbones," because they are monotone; moreover, they are the borders of some important regions in the (b, a)-plane. Hamouly and Mira in 1982 [C. R. Acad. Sc. Paris1 293, 525-528 (1982)] studied the structure of bifurcation of periodic orbits and their mutual position and intersection. Gonchenko et al. [SIAM J. Appl. Dyn. Syst. 4, 407-436 (2005)] display the continuation (in parameter plane) of the first heteroclinic connection and of the first homoclinic connection between the two fixed points of the map. Alligood and Sauer [Commun. Math. Phys. 120, 105-119 (1988)] studied parameter regions characterized by the same rotation number of the "accessible" periodic saddle. Finally, Lorenz [Physica D 237, 1689-1704 (2008)] in 2008 draws areas in the parameter plane statistically characterized by a finite attractor. In this paper, we show how these criteria interact. We therefore conjecture that the wealth of curves of homoclinic onsets could be in general hierarchized by the structure of accessible saddles.

Global bifurcations of a rotating pendulum with irrational nonlinearity

In this paper, the authors consider a rotating pendulum under nonlinear perturbation which allows us to study various kinds of bifurcations and limit cycles. This system exhibits smooth or discontinuous dynamics depending on the value of a mechanical parameter. It is shown that the perturbed smooth system undergoes a pitchfork bifurcation, a homo–heteroclinic orbits transition, a single Hopf bifurcation, a double Hopf bifurcation, a pair of homoclinic bifurcations, a Hopf–homoclinic bifurcation and two saddle–node bifurcation of periodic orbits. The number, position and stability of all the oscillating and rotational limit cycles are given as the parameters vary. We find five limit cycles in the smooth case due to two saddle–node bifurcation of periodic orbits existing. Unlike the smooth case, the perturbed discontinuous system has a homoclinic-like bifurcation and without any saddle–node bifurcation of periodic orbits. Additionally, some results obtained herein bear significant similarities to the codimension-two bifurcation of duffing oscillator and SD oscillator, which may be helpful to explore the codimension bifurcation of the cylindrical pendulum system with irrational nonlinearity.

Multitype Activity Coexistence in an Inertial Two-Neuron System with Multiple Delays

In this paper, an inertial two-neuron system with multiple delays is analyzed to exhibit the effect of time delays on system dynamics. The parameter region with multiple equilibria is obtained employing the pitchfork bifurcation of trivial equilibrium. The stability analysis illustrates that two nontrivial equilibria are both stable for any delays. It implies that the neural system exhibits a stability coexistence of two resting states. Further, due to the existence of multiple delays, the neural system has a periodic activity around the trivial equilibrium via Hopf bifurcation. Finally, numerical simulations are employed to illustrate many richness coexistence for multitype activity patterns. Employing the period-adding route and fold bifurcation of periodic orbit, the neural system may have multistability coexistence of two resting states, two ASP-3s (anti-symmetric periodic activity with period three), one SSP-1 (self-symmetric periodic activity with period one), and one quasi-periodic spiking. Additionally, with increasing delay, quasi-periodic spiking evolves into chaos behavior.

Periodic motion near the surface of asteroids

We are interested in the periodic motion and bifurcations near the surface of an asteroid. The gravity field of an irregular asteroid and the equation of motion of a particle near the surface of an asteroid are studied. The periodic motions around the major body of triple asteroid 216 Kleopatra and the OSIRIS REx mission target asteroid 101955 Bennu are discussed. We find that motion near the surface of an irregular asteroid is quite different from the motion near the surface of a homoplastically spheroidal celestial body. The periodic motions around the asteroid 101955 Bennu and 216 Kleopatra indicate that the geometrical shapes of the orbits are probably very sophisticated. There exist both stable periodic motions and unstable periodic motions near the surface of the same irregular asteroid. This periodic motion which is unstable can be resonant or non resonant. The period doubling bifurcation and pseudo period doubling bifurcation of periodic orbits coexist in the same gravity field of the primary of the triple asteroid 216 Kleopatra. It is found that both of the period doubling bifurcations of periodic orbits and pseudo period-doubling bifurcation of periodic orbits have four different paths. The pseudo period doubling bifurcation found in the potential field of primary of triple asteroid 216 Kleopatra shows that there exist stable periodic orbits near the primary s equatorial plane, which gives an explanation for the motion stability of the triple asteroid 216 Kleopatra s two moonlets, Alexhelios and Cleoselene.

Bifurcations analysis of oscillating hypercycles

We investigate the dynamics and transitions to extinction of hypercycles governed by periodic orbits. For a large enough number of hypercycle species (n>4) the existence of a stable periodic orbit has been previously described, showing an apparent coincidence of the vanishing of the periodic orbit with the value of the replication quality factor Q where two unstable (non-zero) equilibrium points collide (named QSS). It has also been reported that, for values below QSS, the system goes to extinction. In this paper, we use a suitable Poincaré map associated to the hypercycle system to analyze the dynamics in the bistability regime, where both oscillatory dynamics and extinction are possible. The stable periodic orbit is identified, together with an unstable periodic orbit. In particular, we are able to unveil the vanishing mechanism of the oscillatory dynamics: a saddle-node bifurcation of periodic orbits as the replication quality factor, Q, undergoes a critical fidelity threshold, QPO. The identified bifurcation involves the asymptotic extinction of all hypercycle members, since the attractor placed at the origin becomes globally stable for values Q<QPO. Near the bifurcation, these extinction dynamics display a periodic remnant that provides the system with an oscillating delayed transition. Surprisingly, we found that the value of QPO is slightly higher than QSS, thus identifying a gap in the parameter space where the oscillatory dynamics has vanished while the unstable equilibrium points are still present. We also identified a degenerate bifurcation of the unstable periodic orbits for Q=1.

Takens–Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz system

In this work we study Takens–Bogdanov bifurcations of equilibria and periodic orbits in the classical Lorenz system, allowing the parameters to take any real value. First, by computing the corresponding normal form we determine where the Takens–Bogdanov bifurcation of equilibria is non-degenerate, namely of homoclinic or of heteroclinic type. The transition between these two types occurs by means of a triple-zero singularity. Moreover, we demonstrate that a degenerate homoclinic-type Takens–Bogdanov bifurcation of infinite codimension occurs. Secondly, taking advantage of the above analytical results, we carry out a numerical study of the Lorenz system. In this way, we find several kinds of degenerate homoclinic and heteroclinic connections as well as Takens–Bogdanov bifurcations of periodic orbits. The existence of these codimension-two degeneracies, that organize the symmetry-breaking, period-doubling, saddle-node and torus bifurcations undergone by the corresponding periodic orbits, guarantees in some cases the presence of Shilnikov chaos. We also show the existence of a codimension-three homoclinic connection that together with the triple-zero degeneracy act as main organizing centers in the parameter space of the Lorenz system. Finally, we obtain interesting information on the bifurcation sets of the widely studied Chen and Lü systems, taking into account that they are, generically, particular cases of the Lorenz system, as can be proved with a linear scaling in time and state variables. We remark that the heteroclinic case of the Takens–Bogdanov bifurcation in the Lorenz system was found in the literature, in a region with negative parameters, in the study of a thermosolutal convection model and in the analysis of traveling-wave solutions of the Maxwell–Bloch equations.

Bifurcations of periodic solutions and chaos in Josephson system with parametric excitation

Josephson system with parametric excitation is investigated. Using second-order averaging method and Melnikov function, we analyze the existence and bifurcations for harmonic, (2, 3, n-order) subharmonics and (2, 3-order) superharmonics and the heterocilinic and homoclinic bifurcations for chaos under periodic perturbation. Using numerical simulation, we check our theoretical analysis and further study the effect of the parameters on dynamics. We find the complex dynamics, including the jumping behaviors, symmetry-breaking, chaos converting to periodic orbits, interior crisis, non-attracting chaotic set, interlocking (reverse) period-doubling bifurcations from periodic orbits, the processes from interlocking period-doubling bifurcations of periodic orbits to chaos after strange non-chaotic motions when the parameter β increases, etc.

Noose Structure and Bifurcations of Periodic Orbits in Reversible Three-Dimensional Piecewise Linear Differential Systems

The main goal of this work is to describe the periodic behavior of a class of three-dimensional reversible piecewise linear continuous systems. More concretely, we study an interesting structure called the noose bifurcation that was previously detected by Kent and Elgin in the Michelson system. We numerically obtain the curves of periodic orbits that appear from the bifurcations at the noose curve, where other phenomena related to different types of tangencies with the separation plane arise. Besides that, we show that some of these curves of periodic orbits wiggle around global connections when the period increases. The complete structure of periodic orbits, including the stability and bifurcations, coincides with the one observed in the Michelson system. However, we also point out the relevance of the crossing tangency and the small loop that emerges from it in the existence of the noose bifurcation.

Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields

This article deals with the bifurcation of polycycles and limit cycles within the 1-parameter families of planar vector fields Xmk, defined by x˙=y3−x2k+1,y˙=−x+my4k+1, where m is a real parameter and k≥1 is an integer. The bifurcation diagram for the separatrix skeleton of Xmk in function of m is determined and the one for the global phase portraits of (Xm1)m∈R is completed. Furthermore for arbitrary k≥1 some bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of Xmk is found to be uniformly bounded independently of m∈R and the Hilbert number for (Xmk)m∈R, that thus is finite, is found to be at least one.

Topological classifications and bifurcations of periodic orbits in the potential field of highly irregular-shaped celestial bodies

This paper studies the distribution of characteristic multipliers, the structure of submanifolds, the phase diagram, bifurcations and chaotic motions in the potential field of rotating highly irregular-shaped celestial bodies (hereafter called irregular bodies). The topological structure of the submanifolds for the orbits in the potential field of an irregular body is shown to be classified into 34 different cases, including 6 ordinary cases, 3 collisional cases, 3 degenerate real saddle cases, 7 periodic cases, 7 period-doubling cases, 1 periodic and collisional case, 1 periodic and degenerate real saddle case, 1 period-doubling and collisional case, 1 period-doubling and degenerate real saddle case, and 4 periodic and period-doubling cases. The different distribution of the characteristic multipliers has been shown to fix the structure of the submanifolds, the type of orbits, the dynamical behaviour and the phase diagram of the motion. Classifications and properties for each case are presented. Moreover, tangent bifurcations, period-doubling bifurcations, Neimark-Sacker bifurcations and the real saddle bifurcations of periodic orbits in the potential field of an irregular body are discovered. Submanifolds appear to be Mobius strips and Klein bottles when the period-doubling bifurcation occurs.

Continuation of Bifurcations of Periodic Orbits for Large-Scale Systems

A methodology to track bifurcations of periodic orbits in large-scale dissipative systems depending on two parameters is presented. It is based on the application of iterative Newton--Krylov techniques to extended systems. To evaluate the action of the Jacobian it is necessary to integrate variational equations up to second order. It is shown that this is possible by integrating systems of dimension at most four times that of the original equations. In order to check the robustness of the method, the thermal convection of a mixture of two fluids in a rectangular domain has been used as a test problem. Several curves of codimension-one bifurcations, and the boundaries of an Arnold's tongue of rotation number 1/8, have been computed.

Effect of Different Shape of Periodic Forces on Chaotic Oscillation in Brusselator Chemical System

Bifurcations of periodic orbits and chaos in the Brusselator chemical reaction with different shape of external periodic forces are studied numerically. The external periodic forces considered are sine wave, square-wave, rectified sine wave, symmetric saw-tooth wave, asymmetric saw-tooth wave and modulus of sine wave. Period doubling bifurcations, chaos, reverse period doubling bifurcations, quasiperiodic bifurcations are found to occur due to the applied forces. Parametric regimes where suppression of chaos occurs are predicted. Numerical investigations including bifurcation diagram, maximal Lyapunov exponent, Poincare map, Phase portrait and time series plot are used to detect chaos, quasiperiodic and periodic orbits.

Bifurcation analysis of the multiple flips homoclinic orbit

A high-codimension homoclinic bifurcation is considered with one orbit flip and two inclination flips accompanied by resonant principal eigenvalues. A local active coordinate system in a small neighborhood of homoclinic orbit is introduced. By analysis of the bifurcation equation, the authors obtain the conditions when the original flip homoclinic orbit is kept or broken. The existence and the existence regions of several double periodic orbits and one triple periodic orbit bifurcations are proved. Moreover, the complicated homoclinic-doubling bifurcations are found and expressed approximately.

Modulating coherence resonance in non-excitable systems by time-delayed feedback

We propose a paradigmatic model system, a subcritical Hopf normal form subjected to noise and time-delayed feedback, to investigate the impact of time delay on coherence resonance in non-excitable systems. We develop analytical tools to estimate the stationary distribution and the time correlations in nonlinear stochastic delay differential equations. These tools are applied to our model to propose a novel quantity to measure coherence resonance induced by a saddle-node bifurcation of periodic orbits.

1/1 resonant periodic orbits in three dimensional planetary systems

AbstractWe study the dynamics of a two-planet system, which evolves being in a 1/1 mean motion resonance (co-orbital motion) with non-zero mutual inclination. In particular, we examine the existence of bifurcations of periodic orbits from the planar to the spatial case. We find that such bifurcations exist only for planetary mass ratios $\rho=\frac{m_2}{m_1}&lt;0.0205$. For ρ in the interval 0&lt;ρ&lt;0.0205, we compute the generated families of spatial periodic orbits and their linear stability. These spatial families form bridges, which start and end at the same planar family. Along them the mutual planetary inclination varies. We construct maps of dynamical stability and show the existence of regions of regular orbits in phase space.

Cellular mechanisms generating bursting activity in neuronal networks

An open question in neuroscience is how the temporal characteristics of bursting activity are controlled by intrinsic biophysical characteristics. We present two mechanisms organized around the cornerstone bifurcation in a 3D Hodgkin-Huxley style neuronal model. This bifurcation satisfies the criteria for both the Shilnikov blue sky catastrophe and the saddle-node bifurcation on an invariant circle (SNIC) [1-3]. The burst duration (BD) and interburst interval (IBI) increase as the inverse of the square root of the difference between the corresponding parameter and its bifurcation value. The cornerstone bifurcation also determines the stereotypical transient responses of silent and spiking neurons [3]. The mechanisms presented here are based on these transient responses. The first mechanism described half-center oscillator consisting of two intrinsically silent, mutually inhibitory neurons (Figure 1AB). These two neurons were in the silent parameter regime but close to the cornerstone bifurcation. The two coupled neurons showed anti-phase bursting activity without the release and escape mechanisms. We found that if the half-activation voltage of a non-inactivating potassium current (V1/2,mk2) was systematically shifted towards the bifurcation value for the saddle-node bifurcation of periodic orbits, the BDs of both neurons increased in accordance with the inverse-square-root law and exhibited linear dependence on the spike number per burst. Figure 1 The cornerstone bifurcation generates mechanisms controlling bursting in neural networks. (A). Two endogenously silent, mutually inhibitory coupled neurons produce alternating bursting activity. V1/2,mk2 controls BDs. (B). The BDs are depicted as blue ... The second mechanism described the bursting activity of two intrinsically spiking, mutually excitatory neurons. The parameters of the neurons were in vicinity of the cornerstone bifurcation. This network exhibited synchronized bursting (Figure 1CD). Remarkably, excitatory interaction between endogenously spiking neurons essentially led to reduction of excitability of the network. When the half-activation voltage of hyperpolarization-activated current (V1/2,mh) was systematically shifted to the bifurcation value, IBIs of both neurons increased in accordance with the inverse-square-root law and the BDs and the number of spikes per burst stayed constant (6 spikes per burst). This study reveals new mechanisms controlling bursting activity in small neuronal networks based on cellular properties determining transient responses of endogenously spiking and silent neurons. These mechanisms are generic and could govern the bursting regimes in rhythmic neuronal network such as central pattern generators.

Universal unfolding of symmetric resonances

We present the analysis of the bifurcation sequences of a family of resonant 2-DOF Hamiltonian systems invariant under spatial mirror symmetry and time reversion. The phase-space structure is investigated by a singularity theory approach based on the construction of a universal deformation of the detuned Birkhoff–Gustavson normal form. Thresholds for the bifurcations of periodic orbits in generic position are computed as asymptotic series in terms of physical parameters of the original system.