Bifurcations Of Periodic Orbits Research Articles

Sensitivity of the dynamics of the general Rosenzweig-MacArthur model to the mathematical form of the functional response: a bifurcation theory approach.

The equations in the Rosenzweig-MacArthur predator-prey model have been shown to be sensitive to the mathematical form used to model the predator response function even if the forms used have the same basic shape: zero at zero, monotone increasing, concave down, and saturating. Here, we revisit this model to help explain this sensitivity in the case of three response functions of Holling type II form: Monod, Ivlev, and Hyperbolic tangent. We consider both the local and global dynamics and determine the possible bifurcations with respect to variation of the carrying capacity of the prey, a measure of the enrichment of the environment. We give an analytic expression that determines the criticality of the Hopf bifurcation, and prove that although all three forms can give rise to supercritical Hopf bifurcations, only the Trigonometric form can also give rise to subcritical Hopf bifurcation and has a saddle node bifurcation of periodic orbits giving rise to two coexisting limit cycles, providing a counterexample to a conjecture of Kooji and Zegeling. We also revisit the ranking of the functional responses, according to their potential to destabilize the dynamics of the model and show that given data, not only the choice of the functional form, but the choice of the number and/or position of the data points can influence the dynamics predicted.

Bifurcation Analysis of a Rigid Impact Oscillator with Bilinear Damping

This paper studies a rigid impact oscillator with bilinear damping developed as the mechanical model of an impulsive switched system. The stability and the bifurcation of periodic orbits in the impact oscillator are determined by using the mapping methods. One‐parameter bifurcation analyses under variation of forcing frequency and amplitude of external excitation are carried out. Coexisting attractors and various types of bifurcations, such as grazing, period‐doubling, and saddle‐node, are observed, which show the complex phenomena inhered in this impact oscillator.

Bifurcation Gaps in Asymmetric and High-Dimensional Hypercycles

Hypercycles are catalytic systems with cyclic architecture. These systems have been suggested to play a key role in the maintenance and increase of information in prebiotic replicators. It is known that for a large enough number of hypercycle species ([Formula: see text]) the coexistence of all hypercycle members is governed by a stable periodic orbit. Previous research has characterized saddle-node (s-n) bifurcations involving abrupt transitions from stable hypercycles to extinction of all hypercycle members, or, alternatively, involving the outcompetition of the hypercycle by so-called mutant sequences or parasites. Recently, the presence of a bifurcation gap between a s-n bifurcation of periodic orbits and a s-n of fixed points has been described for symmetric five-member hypercycles. This gap was found between the value of the replication quality factor [Formula: see text] from which the periodic orbit vanishes ([Formula: see text]) and the value where two unstable (nonzero) equilibrium points collide ([Formula: see text]). Here, we explore the persistence of this gap considering asymmetries in replication rates in five-member hypercycles as well as considering symmetric, larger hypercycles. Our results indicate that both the asymmetry in Malthusian replication constants and the increase in hypercycle members enlarge the size of this gap. The implications of this phenomenon are discussed in the context of delayed transitions associated to the so-called saddle remnants.

Numerical bifurcation analysis of mathematical models with time delays with the package DDE-BIFTOOL

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and making predictions in various areas of the life sciences, e.g., population dynamics, epidemiology, immunology, physiology, neural networks. The time delays in these models take into account a dependence of the present state of the modeled system on its past history. The delay can be related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period and so on. Due to an infinite-dimensional nature of DDEs, analytical studies of the corresponding mathematical models can only give limited results. Therefore, a numerical analysis is the major way to achieve both a qualitative and quantitative understanding of the model dynamics. A bifurcation analysis of a dynamical system is used to understand how solutions and their stability change as the parameters in the system vary. The package DDE-BIFTOOL is the first general-purpose package for bifurcation analysis of DDEs. This package can be used to compute and analyze the local stability of steady-state (equilibria) and periodic solutions of a given system as well as to study the dependence of these solutions on system parameters via continuation. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. In this paper we describe the structure of DDE-BIFTOOL, numerical methods implemented in the package and we illustrate the use of the package using a certain DDE system.

Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics

Abstract This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from a reduction of symmetric 3D systems with slow-fast dynamics. We concentrate our attention on the saddle dynamics cases, determining the existence of periodic orbits as well as their stability, and possible bifurcations. Dealing with reachable saddles not in the central hysteresis band, we show the existence of subcritical/supercritical heteroclinic bifurcations as well as saddle-node bifurcations of periodic orbits.

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.

Bifurcations of relative periodic orbits in NLS/GP with a triple-well potential

The nonlinear Schrödinger/Gross–Pitaevskii (NLS/GP) equation is considered in the presence of three equally-spaced potentials. The problem is reduced to a finite-dimensional Hamiltonian system by a Galerkin truncation. Families of oscillatory orbits are sought in the neighborhoods of the system’s nine branches of standing wave solutions. Normal forms are computed in the neighborhood of these branches’ various Hamiltonian Hopf and saddle–node bifurcations, showing how the oscillatory orbits change as a parameter is increased. Numerical experiments show agreement between normal form theory and numerical solutions to the reduced system and NLS/GP near the Hamiltonian Hopf bifurcations and some subtle disagreements near the saddle–node bifurcations due to exponentially small terms in the asymptotics.

Dynamical convexity of the Euler problem of two fixed centers

AbstractWe give thorough analysis for the rotation functions of the critical orbits from which one can understand bifurcations of periodic orbits. Moreover, we give explicit formulas of the Conley–Zehnder indices of the interior and exterior collision orbits and show that the universal cover of the regularised energy hypersurface of the Euler problem is dynamically convex for energies below the critical Jacobi energy.

Periodic orbit bifurcations and local symmetry restorations in exotic-shape nuclear mean fields

The semiclassical origins of the enhancement of shell effects in exotic-shape mean-field potentials are investigated by focusing attention on the roles of the local symmetries associated with the periodic-orbit bifurcations. The deformed shell structures for four types of pure octupole shapes in the nuclear mean-field model having a realistic radial dependence are analyzed. Remarkable shell effects are shown for a large Y32 deformation having tetrahedral symmetry. Much stronger shell effects found in the shape parametrization smoothly connecting the sphere and the tetrahedron are investigated from the view-point of the classical–quantum correspondence. The local dynamical symmetries associated with the bridge orbit bifurcations are shown to have significant roles in the emergence of exotic deformed shell structures for certain combinations of the surface diffuseness and the tetrahedral deformation parameters.

Shift dynamics near non-elementary T-points with real eigenvalues

We consider non-elementary T-points in reversible systems in . We assume that the leading eigenvalues are real. We prove the existence of shift dynamics in the unfolding of this T-point. Furthermore, we study local bifurcations of symmetric periodic orbits occurring in the process of dissolution of the chaotic dynamics.

Study of the period function of a two-parameter family of centers

In this paper we study the period function of x¨=(1+x)p−(1+x)q, with p,q∈R and p>q. We prove three independent results. The first one establishes some regions in the parameter space where the corresponding center has a monotonous period function. This result extends the previous ones by Miyamoto and Yagasaki for the case q=1. The second one deals with the bifurcation of critical periodic orbits from the center. The third one is addressed to the critical periodic orbits that bifurcate from the period annulus of each one of the three isochronous centers in the family when perturbed by means of a one-parameter deformation. These three results, together with the ones that we obtained previously on the issue, lead us to propose a conjectural bifurcation diagram for the global behaviour of the period function of the family.

Periodic orbits in tall laterally heated rectangular cavities.

This study elucidates the origin of the multiplicity of stable oscillatory flows detected by time integration in tall rectangular cavities heated from the side. By using continuation techniques for periodic orbits, it is shown that initially unstable branches, arising at Hopf bifurcations of the basic steady flow, become stable after crossing Neimark-Sacker points. There are no saddle-node or pitchfork bifurcations of periodic orbits, which could have been alternative mechanisms of stabilization. According to the symmetries of the system, the orbits are either fixed cycles, which retain at any time the center symmetry of the steady flow, or symmetric cycles involving a time shift in the global invariance of the orbit. The bifurcation points along the branches of periodic flows are determined. By using time integrations, with unstable periodic solutions as initial conditions, we determine which of the bifurcations at the limits of the intervals of stable periodic orbits are sub- or supercritical.

Bifurcations of Cubic Homoclinic Tangencies in Two-dimensional Symplectic Maps

We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic Henon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic Henon maps.

Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model

In this paper, we present an algorithm for deriving the normal forms of Bautin bifurcations in reaction-diffusion systems with time delays and Neumann boundary conditions. On the center manifold near a Bautin bifurcation, the first and second Lyapunov coefficients are calculated explicitly, which completely determine the dynamical behavior near the bifurcation point. As an example, the Segel-Jackson predator-prey model is studied. Near the Bautin bifurcation we find the existence of fold bifurcation of periodic orbits, as well as subcritical and supercritical Hopf bifurcations. Both theoretical and numerical results indicate that solutions with small (large) initial conditions converge to stable periodic orbits (diverge to infinity).

Homoclinic tangencies to resonant saddles and discrete Lorenz attractors

We study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincare) maps and show that the rescaled maps can be brought to a map asymptotically close to the 3D Henon map \begin{document}$\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$\end{document} which, as known [ 14 ], exhibits discrete Lorenz attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of systems possessing discrete Lorenz attractors near the original diffeomorphism.

Centre bifurcations of periodic orbits for some special three dimensional systems

Centre bifurcations of periodic orbits for some special three dimensional systems

Resonance Phenomena in a Scalar Delay Differential Equation with Two State-Dependent Delays

We study a scalar DDE with two delayed feedback terms that depend linearly on the state. The associated constant-delay DDE, obtained by freezing the state dependence, is linear and without recurrent dynamics. With state dependent delay terms, on the other hand, the DDE shows very complicated dynamics. To investigate this, we perform a bifurcation analysis of the system and present its bifurcation diagram in the plane of the two feedback strengths. It is organized by Hopf-Hopf bifurcation points that give rise to curves of torus bifurcation and associated two-frequency dynamics in the form of invariant tori and resonance tongues. We numerically determine the type of the Hopf-Hopf bifurcation points by computing the normal form on the center manifold; this requires the expansion of the functional defining the state-dependent DDE in a power series whose terms up to order three only contain constant delays. We implemented this expansion and the computation of the normal form coefficients in Matlab using symbolic differentiation. Numerical continuation of the torus bifurcation curves confirms the correctness of our normal form calculations. Moreover, it enables us to compute the curves of torus bifurcations more globally, and to find associated curves of saddle-node bifurcations of periodic orbits that bound the resonance tongues. The tori themselves are computed and visualized in a three-dimensional projection, as well as the planar trace of a suitable Poincar\'e section. In particular, we compute periodic orbits on locked tori and their associated unstable manifolds. This allows us to study transitions through resonance tongues and the breakup of a 1:4 locked torus. The work presented here demonstrates that state dependence alone is capable of generating a wealth of dynamical phenomena.

Bifurcation of periodic orbit in planar differential systems with bounded random perturbation

The present paper is dedicated to the bifurcation of periodic orbit for a class of planar differential systems with bounded random perturbation. We first recall three kinds of bifurcations of a nonhyperbolic periodic orbit for planar differential systems with a parameter: Pitchfork bifurcation, saddle-node bifurcation and transcritical bifurcation. The direction and the stability of the periodic orbits bifurcating from the nonhyperbolic periodic orbit are investigated. Then we consider a bounded random perturbation on this system, and study the minimal forward invariant (abbreviated as MFI) sets of the perturbed system. When the parameter changes in a small neighborhood of a bifurcation value, we show that the number of MFI sets changes and the hard bifurcation occurs. Some examples are provided to illustrate our theoretical results.

Analytic Tools to Bound the Criticality at the Outer Boundary of the Period Annulus

In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most $$n\geqslant 0$$ critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family $$\ddot{x}=x^p-x^q,$$ $$p,q\in {\mathbb {R}}$$ with $$p>q$$ .

Shells, orbit bifurcations, and symmetry restorations in Fermi systems

The periodic-orbit theory based on the improved stationary-phase method within the phase-space path integral approach is presented for the semiclassical description of the nuclear shell structure, concerning the main topics of the fruitful activity of V. G. Solovjov. We apply this theory to study bifurcations and symmetry breaking phenomena in a radial power-law potential which is close to the realistic Woods-Saxon one up to about the Fermi energy. Using the realistic parametrization of nuclear shapes we explain the origin of the double-humped fission barrier and the asymmetry in the fission isomer shapes by the bifurcations of periodic orbits. The semiclassical origin of the oblate-prolate shape asymmetry and tetrahedral shapes is also suggested within the improved periodic-orbit approach. The enhancement of shell structures at some surface diffuseness and deformation parameters of such shapes are explained by existence of the simple local bifurcations and new non-local bridge-orbit bifurcations in integrable and partially integrable Fermi systems. We obtained good agreement between the semiclassical and quantum shell-structure components of the level density and energy for several surface diffuseness and deformation parameters of the potentials, including their symmetry breaking and bifurcation values.