Dynamics of one-dimensional granular arrays with pre-compression

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.

Bifurcations analysis of oscillating hypercycles

We study the mechanism of bifurcation of three-dimensional periodic orbits from the plane of motion of the primaries. The branchings of simple and double periodic three-dimensional orbits occur at the plane periodic orbits of indifferent vertical stability. These vertical-critical plane orbits are doubly important. Not only are they necessary in order to specify the vertically stable segments of the families of plane periodic orbits, but they are also the starting points of the families of the simplest possible three-dimensional periodic orbits, namely, the simple and double periodic. We describe here suitable numerical techniques for their accurate determination and for their continuation through the range of the mass parameter μ of the problem. We use these techniques to determine the first few vertical-critical orbits of the basic families of periodic orbits of the problem, namely, families l, m, i, h, a, b and c, for the entire range of μ. The initial conditions of over three hundred vertical-critical orbits are given, classified in twenty-one series. Their horizontal stability is also examined and this leads to predictions of where in the range of μ stable three-dimensional periodic orbits may not branch from these basic families.

In this work we study a wide class of symmetric control systems that has the Chua's circuit as a prototype. Namely, we compute normal forms for Takens–Bogdanov and triple-zero bifurcations in a class of symmetric control systems and determine the local bifurcations that emerge from such degeneracies. The analytical results are used as a first guide to detect numerically several codimension-three global bifurcations that act as organizing centres of the complex dynamics Chua's circuit exhibits in the parameter range considered. A detailed (although partial) bifurcation set in a three'parameter space is presented in this paper. We show relations between several high-codimension bifurcations of equilibria, periodic orbits and global connections. Some of the global bifurcations found have been neither analytically nor numerically treated in the literature.

For differential equations with a compact symmetry group Г some results on global bifurcation of time periodic orbits are presented. In particular it is investigated how the spatial and temporal action of the group Г on a time periodic orbit may vary along global bifurcation branches, e.g. at period doubling bifurcations. The results are obtained geometrically by generic but equivariant approximation, rather than by topological techniques. As an application periodic solutions in reaction diffusion systems with symmetry Dn or O(3) are discussed.

Bifurcation and chaotic behavior of a discrete-time predator–prey system

In this paper we obtain a simple analytic formula for the photodetachment cross section of H 2 in parallel electric and magnetic fields. The three-dimensional semiclassical approximation predicts oscillations in the spectrum and correlates these oscillations with closed classical orbits. The cylindrical symmetry of the Hamiltonian produces some interesting effects. In particular, at boundary energies the semiclassical approximation fails as a focused cusp approaches the origin. @S1050-2947~97!03306-4# PACS number~s!: 32.80.Gc I. INTRODUCTION Experimental measurements of the photodetachment cross section of H 2 in strong static electric fields were made by Bryant et al. @1#. For energies above the threshold energy the resulting cross section was found to be a smooth background upon which was superposed sinusoidal oscillations. Theoretical discussions of the measurements have been given by a number of authors @2,3#. The oscillations arise as an interference effect because the outgoing electron can move against the electric force and then return to the atom. It was shown in Ref. @3# that the cross section could be expressed in the following way: s~ E!5s 0~ E!1C~ E!sinF~ E!, where s 0 is the cross section in the absence of any external fields andC(E) and F(E) are called the recurrence strength and recurrence phase associated with the returning orbit. Theoretical calculations @4,5# show that analogous oscillations occur for photodetachment in crossed electric and magnetic fields, and that the oscillations are again associated with closed orbits. Quantum calculations have shown that very strong oscillations are present in parallel fields @6#, but the relationship to closed orbits was not made clear. Our original motivation in the present paper was to complete this subject by calculating the closed orbits and the resulting spectral oscillations for photodetachment of an electron from H 2 in parallel electric and magnetic fields. This is a system in which the recurrences are simple, strong, relatively easy to observe, and easy to understand. We found results which go far beyond this topic, and which are connected with many other problems of current interest. We know from study of nonlinear dynamics that even when long-time motion is chaotic, short-time motion remains relatively simple and predictable. Periodic orbits play a central role: in Poincare ´ ’s words, they offer ‘‘the only opening through which we might try to penetrate the fortress ~chaos! which has the reputation of being impregnable.’’ This classical statement holds also in quantum mechanics, wherein the periodic-orbit theory of Gutzwiller, Balian and Bloch, and Berry and Tabor @7# provides a general theoretical framework for studying quantum manifestations of classical chaos. Besides producing oscillations in absorption spectra @8#, periodic orbits produce scars in wave functions @9#, oscillations in the density of states @10#, and real-time recurrences that have been observed in many atoms and molecules @11#. @Similar phenomena are observed in microwaves in cavities @12~a!#, in microjunctions @12~b!#, and they are calculated to be consequences of certain models in nuclear physics @12~c!#.# Bifurcations of periodic orbits are of particular interest. A

Classical periodic orbits responsible for emergence of the superdeformed shell structure of single-particle motion in spheroidal cavities are identified and their relative contributions to the shell structure are evaluated. Fourier transforms of quantum spectra clearly show that three-dimensional periodic orbits born out of bifurcations of planar orbits in the equatorial plane become predominant at large prolate deformations. A new semiclassical method capable of describing the shell structure formation associated with these bifurcations is briefly discussed.

Bifurcation of periodic orbits with multiple crossings in a class of planar Filippov systems

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.

We investigate the topological types and bifurcations of periodic orbits in the gravitational field of irregular bodies by the well-known two parameter analysis method. Results show that the topological types of periodic orbits are determined by the locations of these two parameters and that the bifurcation types correspond to their variation paths in the plane. Several new paths corresponding to doubling period bifurcations, tangent bifurcations and Neimark–Sacker bifurcations are discovered. Then, applications in detecting bifurcations of periodic orbits near asteroids 101955 Bennu and 2867 Steins are presented. It is found that tangent bifurcations may occur three times when continuing the vertical orbits near the equilibrium points of 101955 Bennu. The continuation stops as the Jacobi energy reaches a local maximum. However, while continuing the vertical orbits near the equilibrium points of 2867 Steins, the tangent bifurcation and pseudo period-doubling bifurcation occur. The continuation can always go on, and the orbit ultimately becomes nearly circular.

The purpose of this series of lecture notes is to give an outline of the basic tools required to show the occurrence of chaotic motions in the simplest non--integrable problems in Celestial Mechanics, such as the circular restricted planar 3--body problem. No formal proofs will be given here; they can be found in the references given for each section. Section 1 describes the linear and local theory of ordinary differential equations in the neighbourhood of a fixed point; the problems arising in the embedding of invariant stable and unstable manifolds are also discussed. Section 2 is about periodic orbits; the subjects discussed include variational equations, surfaces of section, the continuation of periodic orbits in the restricted 3--body problem, and bifurcation of hyperbolic periodic orbits from resonant periodic orbits. Section 3 covers fundamental models of resonance, the global behaviour of separatrices, and their intersections; all this allows to give at least an outline of the proof of the fundamental result --presented by Poincaré in his book Les méthodes nouvelles de la mécanique céleste-- by which homoclinic points must necessarily occur in the restricted problem. A short conclusion underscores the fact --shown in a rigorous vay much later-- that this in turn implies that chaos in the strongest possible sense occurs in the restricted problem, and is an essential feature of every non--integrable system, even very simple ones with only two degrees of freedom. I apologize for reporting here my lectures in a very short format, almost without comments in between the formulas and statements of the main results; my understanding of the purpose of this notes is that they should serve as a reminder of the existence of many subjects to be studied, rather than a complete presentation which could not be contained in this format.

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.

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.

Compressive deformation and fracture behaviors of AZ31 magnesium alloys with equiaxed grains or bimodal grains