Symmetric Periodic Solutions Research Articles

Dynamics, stability and bifurcations of a planar three-link swimmer with passive visco-elastic joint using “ideal fluid” model

Articulated swimming robots have a promising potential for various marine applications. A common theoretical model assumes ideal fluid, where the viscosity is negligible and the swimmer–fluid interaction is induced by reactive forces originating from added mass effect. Some previous works used this model to study planar multi-link swimmers under kinematic input prescribing all joint angles. Inspired by biological swimmers in nature that utilize body flexibility, in this work we consider an underactuated three-link swimmer where one joint is periodically actuated while the other joint is passive and viscoelastic. Analysis of the swimmer’s nonlinear dynamics reveals that its motion depends significantly on the amplitude and frequency of the actuated joint angle. Optimal frequency is found where the swimmer’s net displacement per cycle is maximized, under symmetric periodic oscillations of the passive joint. In addition, upon crossing critical values of amplitude or frequency, the system undergoes a bifurcation where the symmetric periodic solution loses stability and asymmetric solutions evolve, for which the swimmer moves along an arc. We analyze these phenomena using numerical simulations and analytical methods of perturbation expansion, harmonic balance, Floquet theory, and Hill’s determinant. The results demonstrate the important role of parametric excitation in stability and bifurcations of motion for flexible underactuated locomotion.

A dual principle for symmetric periodic solutions Bi-stability in noninterdigitated Comb-drive MEMS

A dual principle for symmetric periodic solutions Bi-stability in noninterdigitated Comb-drive MEMS

Multiple symmetric periodic solutions of differential systems with distributed delay

In this paper, we study the existence of periodic solutions to the following delay differential system:{x1′(t)=∫−10f1(x2(t+θ))dθ,x2′(t)=∫−10f2(x1(t+θ))dθ. By transforming the problem of searching for periodic solutions of the above system to the problem of finding periodic solutions of an associated ordinary differential system with boundary value conditions and using pseudoindex theory, some sufficient conditions are obtained to guarantee the existence of multiple W-symmetric 2-periodic solutions. We also present three specific examples to illustrate our results.

Periodic Solutions, KAM Tori, and Bifurcations in the Planar Anisotropic Schwarzschild-Type Problem

We investigate analytically the existence of several families of periodic solutions for the planar anisotropic Schwarzschild-type problem. We use reduction and averaging theory, as well as the technique of continuation of Poincaré, for the study of symmetric periodic solutions. Moreover, the determination of KAM 2-tori encasing some of the linearly stable periodic solutions is proved. Finally, we analyze the existence of periodic Hamiltonian pitchfork bifurcation of the periodic solutions.

Second-kind symmetric periodic orbits for planar perturbed Kepler problems and applications

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

The Existence of Odd Symmetric Periodic Solutions in the Generalized Elliptic Sitnikov (N+1)-Body Problem

In this paper, we study the existence of the families of odd symmetric periodic solutions in the generalized elliptic Sitnikov (N+1)-body problem for all values of the eccentricity e∈[0,1) using the global continuation method. First, we obtain the properties of the period of the solution of the corresponding autonomous equation (eccentricity e=0) using elliptic functions. Then, according to these properties and the global continuation method of the zeros of a function depending on one parameter, we derive the existence of odd periodic solutions for all e∈[0,1). It is shown that the temporal frequencies of period solutions depend on the total mass λ (or the number N) of the primaries in a delicate way.

On the stability of symmetric periodic solutions of the generalized elliptic Sitnikov (N + 1)-body problem

In this paper, we study the stability of symmetric periodic solutions of the generalized elliptic Sitnikov (N+1)-body problem. First, based on the relationship between the potential and the period as a function of the energy, we deduce the properties of the period of the solution of the corresponding autonomous equation (eccentricity e=0) in the prescribed energy range. Then, according to these properties and the stability criteria of symmetric periodic solutions of the time-periodic Newtonian equation, we analytically prove the linear stability/instability of the symmetric (m,p)-periodic solutions which emanated from nonconstant periodic solutions of the corresponding autonomous equation when the eccentricity is small, which indicates that the former stability criteria can be extended to the generalized Sitnikov problem with N≥3.

Symmetric periodic solutions of symmetric Hamiltonians in 1 : 1 resonance

The aim of this work is to prove analytically the existence of symmetric periodic solutions of the family of Hamiltonian systems with Hamiltonian function with three real parameters , and . Moreover, we characterize the stability of these periodic solutions as a function of the parameters. Also, we find a first‐order analytical approach of these symmetric periodic solutions. We emphasize that these families of periodic solutions are different from those that exist in the literature.

Minimal [formula omitted]-symmetric periodic solutions of general Hamiltonian systems and delay differential equations

For a skew-symmetric non-degenerate 2n×2n matrix J∈L2n(R) and a J-symplectic matrix M with MTJ−1M=J−1, in this paper we redefine the Maslov-type (J,M)-index for J-symplectic path and develop the iteration theory of the Maslov-type (J,M)-index theory. As applications we study the minimal period problems for M-symmetric orbits of nonlinear autonomous supper-quadratic general Hamiltonian systems, where M is a J-symplectic matrix satisfying Mk=I2n. Also we study the minimal periodic problem for some super-linear delay systems as applications, and we give a positive answer to the Rabinowitz-type minimal period problem for a kind of super-linear delay equations with the forms studied by Kaplan and Yorke in [12].

Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

Symmetric Periodic Solutions for the Spatial Maxwell Restricted $$N+1$$-Problem with Manev Potential

In the spatial Maxwell restricted $$N+1$$ -body problem, the motion of an infinitesimal particle attracted by the gravitational field of (N) bodies is studied. These bodies are arranged in a planar ring configuration. This configuration consists of $$N-1$$ primaries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity $$\omega $$ . Another primary of mass $$m_0=\beta m$$ ( $$\beta >0$$ parameter) is placed at the center of the ring. Moreover, we assume that the central body may be an ellipsoid, or radiation source, which introduces a new parameter e. The existence of several families of symmetric periodic solutions for the spatial Maxwell restricted $$(N+1)$$ -problem with Manev potential is proved. More precisely, firstly we get symmetric periodic solutions around the central body (attractor or repulsor) close to the equatorial plane and small parameter of oblateness. Secondly, we obtain symmetric periodic solutions far away of the central body and peripherals, close to the equatorial plane with arbitrary oblateness. Furthermore, all these families of periodic solutions are stable.

Onset of temporal dynamics within a low reynolds-number laminar fluidic oscillator

We investigate the onset of temporal dynamics of a fluidic oscillator (FO) in the laminar regime at very low Reynolds number (Re=Uhi/ν, where U and hi are the velocity and channel height at inlet, and ν is the kinematic viscosity of the fluid). Both two- and spanwise-periodic-three-dimensional simulations are performed using high-order spectral element methods to characterise the flow inside the FO cavity and the outcoming jet. While the flow remains steady and symmetric for sufficiently low Re, the two-dimensional FO undergoes a symmetry-breaking Hopf bifurcation at ReH2=75.7 that results in a space-time symmetric periodic solution. The remnant space-time symmetry is later broken in a pitchfork bifurcation of limit cycles at ReP2=294.2. Taking three-dimensionality into consideration results in an early spanwise-invariance disruption at about Re3≃68 of wavelength 13hi that retards the onset of the oscillation to ReH3=82.1. This effect is little representative of actual FO implementations, which are typically much narrower than the wavelength of the spanwise instability observed at these low Re. Contrary to what happens with FOs in the turbulent oscillatory regime, the oscillation frequency of the output jet in the laminar regime is found to decrease with Re. The mechanism that drives the oscillation, however, remains the same: as the high speed flow inside the cavity attaches to one of the internal walls through the Coandă effect, the feedback channels divert part of the momentum back, which pushes the incoming flow against the opposite wall. The alternate deviation of the flow inside the cavity to one or the other side results in the periodic flipping of the output jet. The pressure contribution to net momentum at the end of the feedback channels is found to be double that of the advective momentum flux at the early time-periodic regime but both become comparable as Re is increased. The jet sweeping angle amplitude is more pronounced for the two-dimensional FO as compared to three-dimensional at a fixed given Re, the Coandă effect being only partially fulfilled in the latter case. In both cases the sweep amplitude increases with Re. The instability of the output jet, which becomes slightly chaotic already at very low Re, is responsible for triggering the cavity instability that drives the oscillation slightly earlier than it would, should outside noise be suppressed altogether.

Mapping exomoon trajectories around Earth-like exoplanets

ABSTRACT We consider a system in which both the parent star and the Earth-like exoplanet move on circular orbits. Using numerical methods, such as the orbit classification technique, we study all types of trajectories of possible exomoons around the exoplanet. In particular, we scan the phase space around the exoplanet and we distinguish between bounded, collisional, and escaping trajectories, considering both retrograde and prograde types of motion. In the case of bounded regular motion, we also use the grid method and a standard predictor-corrector procedure for revealing the corresponding network of symmetric periodic solutions, while we also compute their linear stability.

A Proof for a Stability Conjecture on Symmetric Periodic Solutions of the Elliptic Sitnikov Problem

In this paper, we aim at proving a conjecture proposed in [X. Cen, X. Cheng, Z. Huang, and M. Zhang, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1271--1290] on the stability of symmetric periodic solu...

Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles

<p style='text-indent:20px;'>In this paper, we prove the existence and multiplicity of subharmonic bouncing motions for a Hill's type sublinear oscillator with an obstacle. Furthermore, we also consider the existence, multiplicity and dense distribution of symmetric periodic bouncing solutions when the weight function is even. Based on an appropriate coordinate transformation and the method of phase-plane analysis, we can study our main results via Poincar<inline-formula><tex-math id="M1">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula> map by applying some suitable fixed point theorems.

Minimal period solutions in asymptotically linear Hamiltonian system with symmetries

<p style='text-indent:20px;'>In this paper, applying the Maslov-type index theory for periodic orbits and brake orbits, we study the minimal period problems in asymptotically linear Hamiltonian systems with different symmetries. For the asymptotically linear semipositive even Hamiltonian systems, we prove that for any given <inline-formula><tex-math id="M1">\begin{document}$ T&gt;0 $\end{document}</tex-math></inline-formula>, there exists a central symmetric periodic solution with minimal period <inline-formula><tex-math id="M2">\begin{document}$ T $\end{document}</tex-math></inline-formula>. Moreover, if the Hamiltonian systems are also reversible, we prove the existence of a central symmetric brake orbit with minimal period being either <inline-formula><tex-math id="M3">\begin{document}$ T $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M4">\begin{document}$ T/3 $\end{document}</tex-math></inline-formula>. Also we give some other lower bound estimations for brake orbits case.

First kind symmetric periodic solutions and their stability for the Kepler problem and anisotropic Kepler problem plus generalized anisotropic perturbation

Motivated by some problems in Celestial Mechanics that combines quasihomogeneous potential in the anisotropic space, we investigate the existence of several families of first kind symmetric periodic solutions for a family of planar perturbed Kepler problem. In addition, we give sufficient conditions for the existence of first kind periodic solutions and also we characterize its type of stability. As an application of this general situation, we discuss the existence of symmetric periodic solutions for the anisotropic Kepler problem plus a generalized anisotropic perturbation, (shortly, p-AKPQ problem) and for the Kepler problem plus a generalized anisotropic perturbation (shortly, p-KPQ problem), as continuation of circular orbits of the two-dimensional Kepler problem. To get this objective, we consider different types of perturbations and then we apply our main result.

Existence of a period two solution of a delay differential equation

We consider the existence of a symmetric periodic solution for the following distributed delay differential equation \begin{document}$ x^{\prime}(t) = -f\left(\int_{0}^{1}x(t-s)ds\right), $\end{document} where \begin{document}$ f(x) = r\sin x $\end{document} with \begin{document}$ r>0 $\end{document} . It is shown that the well studied second order ordinary differential equation, known as the nonlinear pendulum equation, derives a symmetric periodic solution of period \begin{document}$ 2 $\end{document} , expressed in terms of the Jacobi elliptic functions, for the delay differential equation. We here apply the approach in Kaplan and Yorke (1974) for a differential equation with discrete delay to the distributed delay differential equation.

Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

An autonomous Hamiltonian system with two degrees of freedom that is invariant under the Klein four-group of linear canonical automorphisms of the extended phase space of the system is studied. A sequence of symplectic transformations of the monodromy matrix of the symmetric periodic solution of this system is constructed. Using these transformations, the structure and bifurcation of the phase flow in the vicinity of this solution is investigated. It is shown that the bifurcations corresponding to the multiple period increase are different for the solutions with double symmetry and the solutions with a single symmetry. An example of critical periodic solutions of the family of doubly symmetric orbits of the plane circular Hill problem is discussed. The majority of tedious analytical calculations are performed using packages for the computation of Grobner bases and for the work with polynomial ideals in the computer algebra system Maple.

A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

Motivated by the recent discoveries on the stability properties of symmetric periodic solutions of Hamiltonian systems, we establish a Bott-type iteration formula for dihedraly equivariant Hamiltonian systems. We apply the abstract theory for computing the Morse indices of the celebrated Chenciner and Montgomery figure-eight orbit for the planar three body problem in different equivariant spaces. Finally we provide a hyperbolicity criterion for reversible Lagrangian systems.