Existence Of Periodic Motions Research Articles

On Periodic Poincaré Motions in the Case of Degeneracy of an Unperturbed System

This paper is concerned with a one-degree-of-freedom system close to an integrable system. It is assumed that the Hamiltonian function of the system is analytic in all its arguments, its perturbing part is periodic in time, and the unperturbed Hamiltonian function is degenerate. The existence of periodic motions with a period divisible by the period of perturbation is shown by the Poincare methods. An algorithm is presented for constructing them in the form of series (fractional degrees of a small parameter), which is implemented using classical perturbation theory based on the theory of canonical transformations of Hamiltonian systems. The problem of the stability of periodic motions is solved using the Lyapunov methods and KAM theory. The results obtained are applied to the problem of subharmonic oscillations of a pendulum placed on a moving platform in a homogeneous gravitational field. The platform rotates with constant angular velocity about a vertical passing through the suspension point of the pendulum, and simultaneously executes harmonic small-amplitude oscillations along the vertical. Families of subharmonic oscillations of the pendulum are shown and the problem of their Lyapunov stability is solved.

Eigenmodes of Nonlinear Dynamics: Definition, Existence, and Embodiment into Legged Robots with Elastic Elements

Pogo-stick bouncing or the spring loaded inverted pendulum represents fundamental dynamics models for hopping and running in legged locomotion. However, these conceptual models are in general of lower order than the elastic multibody dynamics of versatile segmented legs. The question how to embody these simple models into real robot leg designs still has not been completely answered so far. The concept of eigenmodes for linear systems provides a tool to separate high-dimensional, coupled dynamics in one-dimensional (1-D) invariant ones. However, the dynamics of segmented legs is in general nonlinear such that even the existence of periodic motions, as appearing typically in locomotion tasks, cannot be generally guaranteed without changing intrinsic dynamics behavior substantially by control. This letter extends the concept of eigenmodes, which is well known for linear systems, to the nonlinear case. By proposing a method for selecting the design parameters of multibody systems such that desired eigenmodes are achieved, the problem of embodying fundamental locomotion modes into legged systems is resolved. Examples of practically realizable leg designs are provided, which proof the existence of invariant, 1-D oscillation modes in nonlinear, elastic robot dynamics. An experiment on a multilegged robotic system validates that energetic efficiency can be gained by the proposed approach.

Steady and periodic modes in the problem of motion of a heavy material point on a rotating sphere (the viscous friction case)

The problem of motion of a heavy material point on a sphere uniformly rotating about a fixed axis is considered in the case of viscous friction. The angle of inclination between the axis and the horizon is constant. The existence, bifurcation, and stability of the equilibrium positions are discussed for such a mechanical system. The existence of periodic motions is also studied. An approach is proposed to find such motions in the case of low viscosity.

Bifurcation and Stability of Periodic Motions in a Periodically Forced Oscillator With Multiple Discontinuities

The local stability and existence of periodic motions in a periodically forced oscillator with multiple discontinuities are investigated. The complexity of periodic motions and chaos in such a discontinuous system is often caused by the passability, sliding, and grazing of flows to discontinuous boundaries. Therefore, the corresponding analytical conditions for such singular phenomena to discontinuous boundary are presented from the local singularity theory of discontinuous systems. To develop the mapping structures of periodic motions, basic mappings are introduced, and the sliding motion on the discontinuous boundary is described by a sliding mapping. A generalized mapping structure is presented for all possible periodic motions, and the local stability and bifurcations of periodic motions are discussed. From mapping structures, the switching points of periodic motions on the boundaries are predicted analytically. Two periodic motions are presented for illustrations of the passability, sliding, and grazing of periodic motions on the boundary.

On the existence of periodic motion and the maximum sector bounds for absolute stability of nonlinear nonstationary systems

In this contribution we consider the absolute stability problem. We derive necessary and sufficient conditions for the existence of periodic motion using an operator approach. The results yield an efficient algorithm to approximate the maximum sector bounds for absolute stability numerically.

On maximum sector bounds for absolute stability of single-input single-output systems

We consider the classical problem of absolute stability, i.e. stability of a linear time-invariant system in feedback with a memoryless time-varying sector bound non-linearity. We derive necessary and sufficient conditions for the existence of periodic motion. For the special case of symmetric sector bounds, this condition yields the minimum sector bounds for which the system can sustain periodic motion. By guaranteeing the absence of periodic motion we derive non-conservative sufficient conditions for absolute stability for arbitrary sector bounds. We demonstrate by numerical examples the application of our results.

The existence of periodic motions in rub-impact rotor systems

The existence of rub-impact periodic motions in an eccentric rotor system is considered. A criterion for the periodicity condition of n-periodic impacts is derived and other conditions for real rub-impacts are also discussed. A method consisting of analytical and numerical techniques is presented to solve the existence problem of rub-impact periodic motions. Some special results are obtained by theoretical analyses for rub-impact rotor systems, including the existence of grazing circle motions and that of single-impact periodic motions.

Problems and Techniques

This installment of Problems and Techniques features articles on periodic billiard ball trajectories and the reduction of nearly quadratic polynomial Hamiltonians. Lorenz Halbeisen and Norbert Hungerbühler treat us, in our first article, to a study of periodic billiard ball trajectories on triangular tables. The problem of determining whether or not a billiard ball has a periodic path in a planar domain having a boundary composed of smooth curves has a long history with roots in the eighteenth century. G. D. Birkhoff, for example, established the existence of periodic trajectories on smooth convex domains in 1927. However, the case with nonsmooth boundaries is more difficult, and many situations on polygonal domains are still unresolved. The existence of periodic motion on acute triangles is perhaps the simplest case with motion following the orthoptic triangle with "bounces" at the bases of the three altitudes of the triangle. The existence of periodic trajectories on obtuse triangles is still undetermined and our authors describe cases where periodic motion can be established. They also examine the stability of such motion relative to small perturbations in the domain shape. I don't know if this analysis will improve your game, but it is indeed interesting mathematics. I hope that you enjoy it. In our second article, Jesús Palacián and Patricia Yanguas describe a procedure to reduce the number of degrees of freedom of two-dimensional Hamiltonians that are polynomial functions of the coordinates and momenta with a dominant part that is a homogeneous quadratic function. The popular Fermi--Pasta--Ulam and truncated Toda lattice systems are three-particle systems that may be expressed in this form after a suitable transformation of variables. The Hénon and Heiles family of systems also have the appropriate form. In these circumstances, our authors show how to reduce the number of degrees of freedom of the Hamilitonian system by one. The reduced Hamiltonians are then integrable and the reduced phase space is two-dimensional rather than four-dimensional. Thus, the phase-space diagrams are easily visualized, and this simplifies understanding. The authors characterize the reduced Hamiltonian systems in terms of a collection of normal forms and present conclusions regarding the perturbed Hamiltonian that result from properties of the reduced system. Lots of interesting views of the phase spaces for several reduced Hamiltonians are shown.

Existence of periodic motions of dynamic systems with simple symmetry

Existence of periodic motions of dynamic systems with simple symmetry

Periodic motion in a class of nth-order autonomous differential equations

Sufficient conditions are obtained for the existence of periodic motion in a class of autonomous nonlinear differential equations of order greater than two. The approach is based on the decomposition of an equation into a linear and a nonlinear part. The analysis relies on some basic ideas from linear analysis and geometry. Sufficient conditions for a periodic solution are derived by means of a general topological principle referred to as the torus principle. The existence of a periodic solution is concluded by an appropriate use of the Brouwer fixed-point theorem.

Periodic motion in nonlinear systems

In this paper it is shown how some basic ideas from system theory and differential geometry can be used to establish some new results on the existence of periodic motion in automomous feedback systems. The conditions are expressed in terms of the frequency response characteristic of the open-loop system and certain general properties of the non-linearities.

Oscillation criteria of the Popov type

Criteria of the Popov type are established for the existence of periodic motion in state space for a class of time invariant, autonomous, nonlinear feedback systems. In a restricted sense they constitute an extension of a previously obtained result [1].

Autonomous periodic motion in nonlinear feedback systems

Sufficient conditions are presented for the existence of periodic motions in a class of autonomous, time-invariant, non-linear feedback systems. The conditions can be interpreted as circle type criteria, stated in terms of the frequency response and root locus diagrams for the system's linear part, and in terms of the characteristics of the nonlinearity.

Non-asymptotically stable linear time-invariant systems†

Sufficient conditions for the existence of periodic motion of linear time-invariant systems when represented in phase variable form are obtained through Liapunov's second method. A method of constructing Liapunov functions to obtain conditions for non-asymptotic, non-periodic stability of these systems is suggested. Matrix analogues of these conditions serve to determine the orbital motion and non-asymptotic non-periodic stability of coupled systems whose matrix coefficients are simultaneously diagonalizable.

On periodic motions of stars within ellipsoidal stellar systems

L. PerekThe existence of periodic motions of stars belonging to an ellipsoidal stellar cluster has been proved, the effect of medium resistance being neglected. The density distribution is assumed to depend on the polar semi-axis of the ellipsoidal system, the surfaces of equal density being homothetical ellipsoidal surfaces. By applying Lindstedt–Lyapounov's method of frequency variation the generating solutions have been obtained in an explicit form. Following H. Poincaré's method the first terms of power developments relative to a small parametervhave been found corresponding to several commensurabilities between the mean motion of the star and that of the peri centre of its orbits as well as the angular velocity of rotation of the cluster.