Existence Of Periodic Trajectories Research Articles

Use of Shrink Wrapping for Interval Taylor Models in Algorithms of Computer-Assisted Proof of the Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

Use of Shrink Wrapping for Interval Taylor Models in Algorithms of Computer-Assisted Proof of the Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

Poisson Stability and Periodicity of Control Affine Systems

This article discusses periodic and Poisson stable points of control affine systems. The main result states that the Poisson stability and the periodicity are equivalent concepts for states with closed semi-orbits. This is applied to providing a sufficient condition for the existence of periodic trajectories. Almost periodicity of invariant control systems on Lie groups is specially studied.

Algorithms for Constructing Isolating Sets of Phase Flows and Computer-Assisted Proofs with the Use of Interval Taylor Models

This work continues the research devoted to considering interval Taylor models (TM) as applied to proving the existence of periodic trajectories in systems of ordinary differential equations (ODEs). The Taylor models are used here within a topological approach to constructing an isolating set for the ODE phase flow. We prove auxiliary assertions and propose constructive algorithms for validated numerics over TMs with the aim to expand the domain of their applicability. Necessary topological assertions are proved in order to establish the properties of isolating sets constructed with the aid of TMs. As a result, constructive algorithms are formulated and the main theorem is proved. This theorem makes it possible to construct and verify the homotopy equivalence of the isolating set and the one-dimensional sphere and the homotopy of the mapping of the isolating set into itself to the identity mapping for given systems of ODEs. We also prove the computational complexity of the main algorithm and provide an example of its usage.

Applicability of the Interval Taylor Model to the Computational Proof of Existence of Periodic Trajectories in Systems of Ordinary Differential Equations

We consider the construction of the interval Taylor model used to prove the existence of periodic trajectories in systems of ordinary differential equations. Our model differs from the ones available in the literature in the method for describing the algorithms for the computation of arithmetic operations over Taylor models. In the framework of the current model, this permits reducing the computational expenditures for obtaining interval estimates on computers. We prove an assertion that permits establishing the existence of a periodic solution of a system of ordinary differential equations by verifying the convergence of the Picard iterations in the sense of embedding of the proposed Taylor models. An example illustrating how the resulting assertion can be used to prove the existence of a closed trajectory in the van der Pol system is given.

On Saturn’s rotation relative to a center of mass under the action of the gravitational moments of the Sun and Jupiter

Saturn’s rotation relative to a center of mass is considered within an elliptic restricted three-body problem. It is assumed that Saturn is a solid under the action of gravity of the Sun and Jupiter. The motions of Saturn and Jupiter are considered elliptic with small eccentricities eS and eJ, respectively; the mean motion of Jupiter nJ is also small. We obtain the averaged Hamiltonian function for a small parameter of e = nJ and integrals of evolution equations. The main effects of the influence of Jupiter on Saturn’s rotation are described: (α) the evolution of the constant parameters of regular precession for the angular momentum vector I2; (β) the occurrence of new libration zones of oscillations I2 near the plane of the celestial equator parallel to the plane of the Jupiter’s orbit; (γ) the occurrence of additional unstable equilibria of vector I2 at the points of the north and south poles of the celestial sphere and, as a result, the existence of homoclinic trajectories; and (δ) the existence of periodic trajectories with arbitrarily large periods near the homoclinic trajectory. It is shown that the effects of (β), (γ), and (δ) are caused by the eccentricity e of the Jupiter’s orbit and are practically independent of Jupiter’s mass (within satellite approximation).

Results on the Convergence of Braitenberg Vehicle 3a

Braitenberg vehicles are well-known models of animal behavior used as steering mechanisms in mobile robotics and artificial life. Because of their simplicity, they are mainly used for teaching robotics, while the lack of a quantitative theory has limited their use for research purposes. This article contributes to our formal understanding of Braitenberg vehicle 3a by presenting the convergence properties of its trajectories under parabolic-shaped stimuli. We show previously unreported features of the motion of the vehicle: the conditional stability, the oscillatory behavior, and the existence of periodic trajectories. The mathematical model used provides a theoretical relation between the environment, the internal control mechanism of the vehicle, and some morphological parameters, a link already found in experimental works. This work provides theoretical support for experimental research using Braitenberg vehicle 3a, and paves the way for further research in biology, robotics, and artificial life.

Algorithm for the numerical proof of the existence of periodic trajectories in two-dimensional nonautonomous systems of ordinary differential equations

We suggest an algorithm that permits one to prove the existence of limit periodic trajectories (cycles) in two-dimensional nonautonomous dissipative systems with periodic coefficients with the use of computational methods alone. We prove a fixed point theorem for two-dimensional mappings and describe methods of its application to two-dimensional nonautonomous systems with the use of the Poincare mapping and interval arithmetics.

Stationary Lorentz manifolds and vector fields: existence of periodic trajectories

Stationary Lorentz manifolds and vector fields: existence of periodic trajectories

Periodic solutions for nonautonomous differential equations and inclusions in tubes

We study the existence of periodic trajectories for nonautonomous differential equations and inclusions remaining in a prescribed compact subset of an extended phase space. These sets of constraints are nonconvex right‐continuous tubes not satisfying the viability tangential condition on the whole boundary. We find sufficient conditions for existence of viable periodic trajectories studying properties of the exit subset of the tube. A new approximation approach for continuous multivalued maps is presented.

On Maximal Monotone Differential Inclusions in RN

In this paper we consider differential inclusions driven by a maximalmonotone operator. First we show that for the nonconvex system the solutionset viewed as a multifunction of the initial condition admits a continuousselector passing from a prescribed point. Then we use this selector to showthe path connectedness of the solution set. We also investigate thecontinuity properties of the solution multifunction. Finally we solve aviability problem and we also establish the existence of periodic trajectories.

On Periodic Solutions of Multivalued Nonlinear Evolution Equations

In this paper first we prove a new existence theorem for periodic solutions of finite dimensional differential inclusions. Then we use this result and Galerkin approximations to establish the existence of periodic trajectories for a class of nonlinear evolution inclusions. An example of a parabolic nonlinear feedback control system is also worked out in detail.

Periodic bounce solutions of dynamical conservative systems and their minimal period

SEVERAL important studies have been made on the problem of finding periodic solutions of a conservative dynamical system. An important part of such a study is concerned with the case in which the minimal period of the trajectory is given (see e.g. [l, 3, 6, 9, 14)). In this paper we consider the problem of existence of periodic trajectories with prescribed minimal period, when the material point moves in a convex billiard in I?” under the action of a conservative field and it bounces in a completely elastic way against the boundary, hitting it orthogonally. If Q is a bounded open convex subset of I?” (not necessarily regular), T > 0 and V is a real Cz-function in a neighbourhood of 0, we consider the nonconstant periodic orbits q : R” -+ Q (which are called “principal bounce trajectories” @.b.t.)) having period 2T such that:

On the problem op n-stationary centers

Among the various problems of celestial mechanics related to the n-body problem, a certain amount of interest attaches to the specific situation wherein a passive gravitational point mass M moves under the assumption that the relative disposition of the other active gravitational masses experiences no large changes. If the attracting masses are regarded as points and if changes in the relative disposition of the attracting bodies are neglected, one arrives at the problem of the motion of the point M in a field produced by n-stationary attracting centers (the point M here represents the ( n+ l)-th body). In addition to the problem of central motion ( n = 1), soluble dynamics problems of this category include Euler's case [1] of two ( n= 2) stationary Newtonian attracting centers. This problem, which for a long time was of solely theoretical Interest as an example of an integrable Liouville system [2], has recently been attracting attention in connection with the mechanics of artificial satellites, particularly after it was shown that the potential of an oblate spheroid can be approximated by the potential of two specifically chosen stationary Newtonian attracting centers [3 and 4]. The solution of the problem for n-attracting centers for n ≥ 3 is unknown, except for a single special case of three centers pointed out by Lagrange and considered In greater detail by J.A. Serre [5]. We shall concern ourselves here with problems on the existence of periodic trajectories in the case of n-attracting centers. An arbitrary and not necessarily Newtonian gravitational law will be assumed. Our analysis is based on the theory of quasiindices of singular force field points as set forth in [60].