Periodic solutions of linear impulsive differential inclusions

We establish sufficient conditions for the existence of periodic R-solutions of linear differential inclusions with impulses at fixed times. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1287–1296, September, 2008.     

Periodic solutions of delay impulsive differential equations

We study the following semilinear impulsive differential equation with delay:

     

Periodic solutions to impulsive differential inclusions with constraints

The existence of a periodic solution to an impulsive differential inclusion being invariant with respect to a non-convex set of state constraints is established by the use a Lefschetz type fixed-point theorem for set-valued maps.     

Periodic and almost periodic solutions of strongly non-linear impulsive systems

The problem of the existence and stability of periodic and almost periodic solutions of strongly non-linear impulsive systems is investigated. The Poincaré method [1] is justified for the case of an isolated generating solution. A dynamical system consisting of a bead on a vibrating surface is considered as an example.

The small parameter method for investigating systems with discontinuous solutions was previously applied [2, 3] to the case when the periodic solution is non-isolated.

A method is used below for reducing the investigation of a system of equations with impulsive actions on surfaces to equations with fixed moments of inpulsive action.     

Periodic solutions of nonlinear impulsive differential inclusions with constraints

In this paper we obtain the existence of periodic solutions for nonlinear ``invariance' problems monitored by impulsive differential inclusions subject to impulse effects.

     

Periodic solutions for impulsive semi-ratio-dependent predator–prey systems

This paper investigates the existence of periodic solutions for an impulsive semi-ratio-dependent predator–prey system with most popular prey-dependent (monotone or non-monotone) and predator-dependent functional response. Sharp sufficient conditions are derived by the invariance property of homotopy and analysis technique. The presented criteria improve and extend many previous results in the literature.     

Uncertain inference control for balancing an inverted pendulum

Fuzzy inference control uses fuzzy sets to describe the antecedents and consequents of If-Then rules. However, most surveys show the antecedents and consequents are uncertain sets rather than fuzzy sets. This fact provides a motivation to invent an uncertain inference control method. This paper gives an introduction to the design procedures of uncertain inference controller. As an example, an uncertain inference controller for balancing an inverted pendulum system is successfully designed. The computer simulation shows the developed uncertain inference controller is of good robustness.     

Nonlinear boundary-value problems for Boltzmann equations: Periodic solutions and their bifurcations

The statement of nonlinear boundary conditions for the Boltzmann kinetic equation is examined. On this basis, the possibility is discussed that relaxation- or preturbulent-type oscillations that are asymptotically periodic can form in gases. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 323–333, February, 1997.     

Periodic solutions for ordinary differential equations with sublinear impulsive effects

The continuation method of topological degree is used to investigate the existence of periodic solutions for ordinary differential equations with sublinear impulsive effects. The applications of the abstract approach include the generalizations of some classical nonresonance theorem for impulsive equations, for instance, the existence theorem for asymptotically positively homogeneous differential systems and the existence theorem for second order equations with Landesman-Lazer conditions.     

Deterministic chaos of a spherical pendulum under limited excitation

We investigate the appearance, development, and vanishing of deterministic chaos in a “spherical pendulum-electric motor of limited power” dynamical system. Chaotic attractors discovered in the system are described in detail. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 4, pp. 534–548, April, 2007.     

Parametric excitation of an internally resonant double pendulum

Summary A system of two weakly damped, nonlinearly coupled pendulums with (uncoupled) natural frequencies of ₁ and ₂ 2 ₁ is subjected to a small, vertical oscillation of frequency 22 ₁. The resulting motion (after transients have decayed) may be either a simple (rigid-body) translation of the entire system or coupled oscillations of the pendulums superimposed on such a translation. The stability of these two types of motion is examined and their parametric domains established. There are no Hopf bifurcations, in consequence of which neither periodically modulated nor chaotic motions appear to be possible (for weak excitation). The corresponding problem for a symmetric triple pendulum is shown to be isomorphic to that for the double pendulum.

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Zusammenfassung Ein System von schwach gedämpften, nichtlinear gekoppelten Pendeln mit den (ungekoppelten) natürlichen Frequenzen ₁ und ₂2 ₁ wird vertikalen erzwungenen Schwingungen mit der Frequenz 22 ₁ unterworfen. Die entstehende Bewegung ist (nach dem Abklingen von temporären Bewegungsanteilen) entweder eine einfache Translation des (starren) Systems oder aber eine solche Translation mit überlagerten gekoppelten Schwingungen der Pendel. Die Stabilität dieser beiden Bewegungstypen wird untersucht, und ihr parametrischer Gültigkeitsbereich wird bestimmt. Es gibt keine Hopf-Bifurkationen, so daß weder periodisch modulierte noch chaotische Bewegungen möglich erscheinen (für schwache Erregungen). Das entsprechende Problem für ein symmetrisches Dreifach-Pendel ergibt sich isomorph zu dem des Doppel-Pendels.

     

The stability of an inverted pendulum with a vibrating suspension point

The problem of stabilizing the upper vertical (inverted) position of a pendulum using vibration of the suspension point is considered. The periodic function describing the vibrations of the suspension point is assumed to be arbitrary but possessing small amplitudes, and slight viscous damping is taken into account. A formula is obtained for the limit of the region of stability of the solutions of Hill's equation with damping in the neighbourhood of the zeroth natural frequency. The analytical and numerical results are compared and show good agreement. An asymptotic formula is derived for the critical stabilization frequency of the upper vertical position of the pendulum. It is shown that the effect of viscous damping on the critical frequency is of the third-order of smallness and, in all the examples considered, when viscous damping is taken into account the critical frequency increases.     

Expansion of solution of an inverted pendulum system with time delay

In this paper, the solution expansion for an inverted pendulum system with time delay is studied. The linearized model of this nonlinear system near its equilibrium is derived on the assumption that a unique equilibrium exists in it. Then the asymptotic expressions of its eigenvalues and the eigenvalues’ corresponding eigenvectors are obtained. Moreover, although the set of these eigenvectors does not form a Schauder basis for the state space, the solution of this model still can be expressed by these eigenvectors in the form of infinite series under certain conditions. Finally, a simulation is provided to support these results.     

Parametric excitation of an internally resonant double pendulum,II

Summary The small, finite amplitude response of a damped, internally resonant double pendulum subject to parametric excitation at four times the frequency of the dominant pendulum is shown to exhibit chaotic behavoir for certain values of frequency offset and dissipation. This is in contrast to Miles [1], where the driving frequency approximated twice that of the dominant pendulum, and regular motion was found to exist for all values of frequency offset and dissipation.

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Zusammenfassung Es wird gezeigt, daß das Verhalten kleiner, endlicher Amplitude eines gedämpften, intern resonanten Doppelpendels unter parametrischer Erregung mit der vierfachen Eigenfrequenz des Hauptpendels chaotisch ist für bestimmte Frequenz- und Dissipationswerte. Dies steht im Gegensatz zu Miles [1], wo die erregende Frequenz ungefähr das zweifache der Eigenfrequenz betrug und sich normale Bewegung für alle Frequenzen und Dissipationen ergab.

     

Periodic motions and bifurcations of a vibro-impact system

Based on the analysis of a two-degree-of-freedom plastic impact oscillator, we introduce a three-dimensional map with dynamical variables defined at the impact instants. The non-linear dynamics of the vibro-impact system is analyzed by using the Poincaré map, in which piecewise property and singularity are found to exist. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after the impact, and the singularity of map is generated via the grazing contact of two masses and corresponding instability of periodic motions. These properties of the map have been shown to exhibit particular types of sliding and grazing bifurcations of periodic-impact motions under parameter variation. Simulations of the free flight and sticking solutions are carried out, and regions of existence and stability of different impact motions are therefore presented in (δ, ω) plane of dimensionless clearance δ and frequency ω. The influence of non-standard bifurcations on dynamics of the vibro-impact system is elucidated accordingly.     

Diffusion dynamics near critical bifurcations in a nonlinearly damped pendulum system

We numerically study the diffusion dynamics near critical bifurcations such as sudden widening of the size of a chaotic attractor, intermittency and band-merging of a chaotic attractor in a nonlinearly damped and periodically driven pendulum system. The system is found to show only normal diffusion. Near sudden widening and intermittency crisis power-law variation of diffusion constant with the control parameter ω of the external periodic force f sin ωt is found while linear variation of it is observed near band-merging crisis. The value of the exponent in the power-law relation varies with the damping coefficient and the strength of the added Gaussian white noise.