Continuation Of Periodic Orbits Research Articles

On the periodic orbits of the continuous–discontinuous piecewise differential systems with three pieces separated by two parallel straight lines

During these last twenty years many papers have been published about the piecewise differential systems in the plane. The increasing interest on these kind of differential systems mainly is due to their big number of applications for modeling many natural phenomena. As usual one of the main difficulties for understanding the dynamics of the differential systems in the plane consists in controlling their periodic orbits and mainly their limit cycles. Therefore there are a big number of papers studying the existence or non-existence of periodic orbits of the continuous and discontinuous piecewise differential systems, but as far as we know this paper will be one of the first papers studying the periodic orbits of a class of piecewise differential systems such that in a part of the line of separation between the two differential systems forming the piecewise differential system is continuous and in the other part it is discontinuous.Thus we shall study the piecewise differential systems separated by two parallel straight lines, in one of these straight lines the piecewise differential system is continuous and in the other discontinuous, moreover in the three pieces of these piecewise differential systems we put arbitrary Hamiltonian systems of degree one. We obtain that such kind of continuous–discontinuous piecewise differential systems cannot have limit cycles, but they can have a continuum of periodic orbits.

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

Nonlinear dynamic analysis and numerical continuation of periodic orbits in high-index differential–algebraic equation systems

Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré

We present an extension of a classical result of Poincare (1892) about continuation of periodic orbits and breaking of completely resonant tori in a class of nearly integrable Hamiltonian systems, which covers most Hamiltonian Lattice models. The result is based on the fixed point method of the period map and exploits a standard perturbation expansion of the solution with respect to a small parameter. Two different statements are given, about existence and linear stability: a first one, in the so called non-degenerate case, and a second one, in the completely degenerate case. A pair of examples inspired to the existence of localized solutions in the discrete NLS lattice is provided.

A Semi-Analytical Method for Calculation of Strongly Nonlinear Normal Modes of Mechanical Systems

A new scheme based on the homotopy analysis method (HAM) is developed for calculating the nonlinear normal modes (NNMs) of multi degrees-of-freedom (MDOF) oscillatory systems with quadratic and cubic nonlinearities. The NNMs in the presence of internal resonances can also be computed by the proposed method. The method starts by approximating the solution at the zeroth-order, using some few harmonics, and proceeds to higher orders to improve the approximation by automatically including higher harmonics. The capabilities and limitations of the method are thoroughly investigated by applying them to three nonlinear systems with different nonlinear behaviors. These include a two degrees-of-freedom (2DOF) system with cubic nonlinearities and one-to-three internal resonance that occurs on nonlinear frequencies at high amplitudes, a 2DOF system with quadratic and cubic nonlinearities having one-to-two internal resonance, and the discretized equations of motion of a cylindrical shell. The later one has internal resonance of one-to-one. Moreover, it has the symmetry property and its DOFs may oscillate with phase difference of 90 deg, leading to the traveling wave mode. In most cases, the estimated backbone curves are compared by the numerical solutions obtained by continuation of periodic orbits. The method is found to be accurate for reasonably high amplitude vibration especially when only cubic nonlinearities are present.

Limit Cycles of Piecewise Smooth Differential Equations on Two Dimensional Torus

In this paper we study the limit cycles of some classes of piecewise smooth vector fields defined in the two dimensional torus. The piecewise smooth vector fields that we consider are composed by linear, Ricatti with constant coefficients and perturbations of these one, which are given in (3). Considering these piecewise smooth vector fields we characterize the global dynamics, studying the upper bound of number of limit cycles, the existence of non-trivial recurrence and a continuum of periodic orbits. We also present a family of piecewise smooth vector fields that posses a finite number of fold points and, for this family we prove that for any 2k number of limit cycles there exists a piecewise smooth vector fields in this family that presents k number of limit cycles and prove that some classes of piecewise smooth vector fields presents a non-trivial recurrence or a continuum of periodic orbits.

Continuation of periodic orbits in the Sun-Mercury elliptic restricted three-body problem

Starting from resonant Halo orbits in the Circular Restricted Three-Body Problem (CRTBP), Multi-revolution Elliptic Halo (ME-Halo) orbits around L1 and L2 points in the Sun-Mercury Elliptic Restricted Three-Body Problem (ERTBP) are generated systematically. Three pairs of resonant parameters M5N2, M7N3 and M9N4 are tested. The first pair shows special features and is investigated in detail. Three separated characteristic curves of periodic orbit around each libration point are obtained, showing the eccentricity varies non-monotonically along these curves. The eccentricity of the Sun-Mercury system can be achieved by continuation method in just a few cases. The stability analysis shows that these orbits are all unstable and the complex instability occurs with certain parameters. This paper shows new periodic orbits in both the CRTBP and the ERTBP. Totally four periodic orbits with parameters M5N2 around each libration points are extracted in the Sun-Mercury ERTBP.

PERIODIC ORBIT FAMILIES IN THE GRAVITATIONAL FIELD OF IRREGULAR-SHAPED BODIES

ABSTRACT The discovery of binary and triple asteroids in addition to the execution of space missions to minor celestial bodies in the past several years have focused increasing attention on periodic orbits around irregular-shaped celestial bodies. In the present work, we adopt a polyhedron shape model for providing an accurate representation of irregular-shaped bodies and employ the model to calculate their corresponding gravitational and effective potentials. We also investigate the characteristics of periodic orbit families and the continuation of periodic orbits. We prove a fact, which provides a conserved quantity that permits restricting the number of periodic orbits in a fixed energy curved surface about an irregular-shaped body. The collisions of Floquet multipliers are maintained during the continuation of periodic orbits around the comet 1P/Halley. Multiple bifurcations in the periodic orbit families about irregular-shaped bodies are also discussed. Three bifurcations in the periodic orbit family have been found around the asteroid 216 Kleopatra, which include two real saddle bifurcations and one period-doubling bifurcation.

Center boundaries for planar piecewise-smooth differential equations with two zones

This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary. We prove that given a pair of systems that share a hyperbolic focus singularity p0, with the same orientation and opposite stability, and a ray Σ0 with endpoint at the singularity p0, we can find a smooth manifold Ω such that Σ0∪{p0}∪Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached.

Canard solutions in planar piecewise linear systems with three zones

ABSTRACTIn this work, we analyze the existence and stability of canard solutions in a class of planar piecewise linear systems with three zones, using a singular perturbation theory approach. To this aim, we follow the analysis of the classical canard phenomenon in smooth planar slow–fast systems and adapt it to the piecewise-linear framework. We first prove the existence of an intersection between repelling and attracting slow manifolds, which defines a maximal canard, in a non-generic system of the class having a continuum of periodic orbits. Then, we perturb this situation and we prove the persistence of the maximal canard solution, as well as the existence of a family of canard limit cycles in this class of systems. Similarities and differences between the piecewise linear case and the smooth one are highlighted.

Continuation of periodic orbits in symmetric Hamiltonian and conservative systems

We present and review results on the continuation and bifurcation of periodic solutions in conservative, reversible and Hamiltonian systems in the presence of symmetries. In particular we show how two-point boundary value problem continuation software can be used to compute families of periodic solutions of symmetric Hamiltonian systems. The technique is introduced with a very simple model example (the mathematical pendulum), justified with a theoretical continuation result and then applied to two non trivial examples: the non integrable spring pendulum and the continuation of the figure eight solution of the three body problem.

Symmetric Periodic Orbits and Schubart Orbits in The Charged Collinear Three-Body Problem

Using analytic continuation of periodic orbits and symmetries we construct symmetric periodic solutions in the charged collinear three-body problem for the case where the outer particles are sufficiently small and have small charge. The results obtained allows to construct Schubart-like periodic solutions for different values of masses and charges in this setting and for the case of equal masses and charges of the outer bodies we have explicit Schubart periodic solutions.

Melnikov theory for a class of planar hybrid systems

In this paper, we extend the well-known Melnikov theory for smooth systems to a class of two-zonal planar hybrid piecewise smooth systems. In this class, each zone of differentiability is separated by a straight line and in these zones, the dynamics are governed by a smooth system. When an orbit reaches the separation line then a reset map applies before entering the orbit in the other zone. The family of non-generic systems of this class which has a continuum of periodic orbits is considered. Then, we study the periodic orbits of the continuum that persist under a perturbation. To achieve this objective, first we build an appropriate Poincaré map, after that, we introduce a function of one variable whose zeros provide us the periodic orbits that remain after the perturbation. This function will be called the Melnikov function because it is very close to the so-called Melnikov function for smooth systems. Finally, we apply the obtained results to some piecewise linear systems, continuous as well as discontinuous ones.

On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems

Numerical computation of nonlinear normal modes (NNMs) for conservative vibratory systems is addressed, with the aim of deriving accurate reduced-order models up to large amplitudes. A numerical method is developed, based on the center manifold approach for NNMs, which uses an interpretation of the equations as a transport problem, coupled to a periodicity condition for ensuring manifold's continuity. Systematic comparisons are drawn with other numerical methods, and especially with continuation of periodic orbits, taken as reference solutions. Three different mechanical systems, displaying peculiar characteristics allowing for a general view of the performance of the methods for vibratory systems, are selected. Numerical results show that invariant manifolds encounter folding points at large amplitude, generically (but not only) due to internal resonances. These folding points involve an intrinsic limitation to reduced-order models based on the centre manifold and on the idea of a functional relationship between slave and master coordinates. Below that amplitude limit, numerical methods are able to produce reduced-order models allowing for a precise prediction of the backbone curve.

CONNECTING SYMMETRIC AND ASYMMETRIC FAMILIES OF PERIODIC ORBITS IN SQUARED SYMMETRIC HAMILTONIANS

In this work, we study a generic squared symmetric Hamiltonian of two degrees of freedom. Our aim is to show a global methodology to analyze the evolution of the families of periodic orbits and their bifurcations. To achieve it, we use several numerical techniques such as a systematic grid search algorithm in sequential and parallel, a fast chaos indicator and a tool for the continuation of periodic orbits. Using them, we are able to study the special and generic bifurcations of multiplicity one that allow us to understand the dynamics of the system and we show in detail the evolution of some symmetric breaking periodic orbits.

Saddle–node bifurcation of invariant cones in 3D piecewise linear systems

In this work, the existence of a saddle–node bifurcation of invariant cones in three-dimensional continuous homogeneous piecewise linear systems is considered. First, we prove that invariant cones for this class of systems correspond one-to-one to periodic orbits of a continuous piecewise cubic system defined on the unit sphere. Second, let us give the conditions for which the sphere is foliated by a continuum of periodic orbits. The principal idea is looking for the periodic orbits of the continuum that persist when this situation is perturbed. To do this, we establish the relationship between the invariant cones of the three-dimensional system and the periodic orbits of two planar hybrid piecewise linear systems. Next, we define two functions whose zeros provide the invariant cones that persist under the perturbation. These functions will be called Melnikov functions and their properties allow us to state some results about the existence of invariant cones and other results about the existence of saddle–node bifurcations of invariant cones, which is the principal goal of this paper.

ON THE BIFURCATION AND CONTINUATION OF PERIODIC ORBITS IN THE THREE BODY PROBLEM

We consider the planar three body problem of planetary type and we study the generation and continuation of periodic orbits and mainly of asymmetric periodic orbits. Asymmetric orbits exist in the restricted circular three body problem only in particular resonances called "asymmetric resonances". However, numerical studies showed that in the general three body problem, asymmetric orbits may exist not only for asymmetric resonances, but for other kinds, too. In this work, we show the existence of asymmetric periodic orbits in the elliptic restricted problem. These families of periodic orbits continue existing and clarify the origin of many asymmetric periodic orbits in the general problem. Also, we illustrate how the families of periodic orbits of the restricted circular problem and those of the elliptic one join smoothly and form families in the general problem, verifying in this way the scenario first described by Bozis and Hadjidemetriou.

Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems

We study relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. The system consists of two identical heavy atoms and a light one, and the atomic mass ratio is treated as a continuation parameter. Under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists for sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). The proof is based on a method initially introduced by Sepulchre and MacKay [J.-A. Sepulchre, R.S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713] and further developed by Muñoz-Almaraz et al. [F.J. Muñoz-Almaraz, et al., Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D 181 (2003) 1–38] for the continuation of normal periodic orbits in Hamiltonian systems. Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes [J.-P. Ortega, Relative normal modes for nonlinear Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665–704] up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms. This analysis is completed by numerical results on the stability and bifurcations of some inversion orbits as their angular velocity is varied.

Lower bounds for the number of limit cycles of trigonometric Abel equations

We consider the Abel equation x ˙ = A ( t ) x 3 + B ( t ) x 2 , where A ( t ) and B ( t ) are trigonometric polynomials of degree n and m, respectively, and we give lower bounds for its number of isolated periodic orbits for some values of n and m. These lower bounds are obtained by two different methods: the study of the perturbations of some Abel equations having a continuum of periodic orbits and the Hopf-type bifurcation of periodic orbits from the solution x = 0 .

Continuation of periodic orbits in a ZAD-strategy controlled buck converter

This paper concerns the dynamics of a Zero Average Dynamics (ZAD) controlled DC–DC Buck converter. We study the continuation problem of periodic orbits in a periodically forced piecewise-smooth system through the ranges of existence and stability. These orbits can have different configuration and periodicity, and they end in a transition to chaotic bands when a parameter is varied. Three assumptions (a symmetry assumption, a zero-average assumption and a regulation assumption) allows existence ranges to be predicted analytically, and there is only a final efficient numerical step. Stability is checked through Floquet exponents, which are also analytically computed.

Continuation of Periodic Orbits of a Third Order Oscillator under Autonomous Perturbation

We consider systems of nonlinear third-order differential equations depending on a small parameter such that the unperturbed system has a three dimension manifold of periodic solutions. We impose some conditions on the system in order to reduce the third order perturbed system to the second one. By using Poincare-Andronov-Hopf theorem we prove the existence of many periodic solutions the perturbed system.