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Ivan Polekhin

Publications and source records attributed to Ivan Polekhin.

17 recordsLinked to original sources
arXiv2024

On the dynamics and integrability of the Ziegler pendulum

We prove that the Ziegler pendulum -- a double pendulum with a follower force -- can be integrable, provided that the stiffness of the elastic spring located at the pivot point of the pendulum is zero and there is no friction in the system. We show that the integrability of the system follows from the existence of two-parameter families of periodic solutions. We explain a mechanism for the transition from integrable dynamics, for which there exist two first integrals and solutions belong to two-dimensional tori in a four-dimensional phase space, to more complicated dynamics. The case in which the stiffnesses of both springs are non-zero is briefly studied numerically. We show that regular dynamics coexists with chaotic dynamics.

math.DS↗
arXiv2022

Asymptotically Stable Non-Falling Solutions of the Kapitza-Whitney Pendulum

The planar inverted pendulum with a vibrating pivot point in the presence of an additional horizontal force field is studied. The horizontal force is not assumed to be small or rapidly oscillating. We assume that the pivot point of the pendulum rapidly oscillates in the vertical direction and the period of these oscillations is commensurable with the period of horizontal force. This system can be considered as a strong generalization of the Kapitza pendulum. Previously it was shown that for any horizontal force there always exists a non-falling periodic solution in the considered system. In particular, when there is no horizontal force, this periodic solution is the vertical upward position. In the paper we present analytical and numerical results concerning the existence of asymptotically stable non-falling periodic solutions in the system.

math.DS↗
arXiv2021

The Spherical Kapitza-Whitney Pendulum

In this paper we study the global dynamics of the inverted spherical pendulum with a vertically vibrating suspension point in the presence of an external horizontal periodic force field. We do not assume that this force field is weak or rapidly oscillating. We prove that there always exists a non-falling periodic solution, i.e., there exists an initial condition such that the rod of the pendulum never becomes horizontal along the corresponding solution. We also show numerically that there exists an asymptotically stable non-falling solution for a wide range of parameters of the system.

math.CA↗
arXiv2021

On the Application of the Ważewski Method to the Problem of Global Stabilization

We consider a possible application of the Ważewski topological method to feedback control systems and to more general dynamical systems. We show how this method can be used to prove the impossibility of global stabilization in such problems. Moreover, we give sufficient conditions for the existence of a solution such that its trajectory never leaves a subset of the extended phase space of the system and does not tend asymptotically to a given equilibrium. We illustrate our result with various real-life systems including the Furuta pendulum and the wheeled inverted pendulum.

math.OC↗
arXiv2020

On the non-integrability and dynamics of discrete models of threads

In the paper, we study the dynamics of planar $n$-gons, which can be considered as discrete models of threads. The main result of the paper is that, under some weak assumptions, these systems are not integrable in the sense of Liouville. This holds for both completely free threads and for threads with fixed points that are placed in external force fields. We present sufficient conditions for the positivity of topological entropy in such systems. We briefly consider other dynamical properties of discrete threads and we also consider discrete models of inextensible yet compressible threads.

nlin.CD↗
arXiv2020

The method of averaging for the Kapitza-Whitney pendulum

A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic horizontal force field. We study the existence of forced oscillations in the system. It is shown that there always exists a periodic solution along which the rod of the pendulum never becomes horizontal, i.e. the pendulum never falls, provided the period of vibration and the period of horizontal force are commensurable. We also present a sufficient condition for the existence of at least two different periodic solutions without falling. We show numerically that there exist stable periodic solutions without falling.

math.DS↗
arXiv2019

Averaging method and asymptotic solutions in some mechanical problems

In the paper we consider systems in oscillating force fields such that the classical method of averaging can be applied. We present sufficient conditions for the existence of forced oscillations in such systems and study the asymptotic behaviour of some solutions. In particular, we show that for an inverted pendulum with a horizontally moving pivot point in an oscillating gravity field there exists a solution along which the pendulum never falls.

math.DS↗
arXiv2019

Remarks on forced oscillations in some systems with gyroscopic forces

In the paper we study the existence of a forced oscillation in two Lagrange systems with gyroscopic forces: a spherical pendulum in a magnetic field and a point on a rotating closed convex surface. We show how it is possible to prove the existence of forced oscillations in these systems provided the systems move in the presence of viscous friction.

math.DS↗
arXiv2018

Precession of the Kovalevskaya and Goryachev-Chaplygin tops

The change of the precession angle is studied analytically and numerically for the integrable tops of Kovalevskaya and Goryachev-Chaplygin. Based on the known results on the topology of Liouville foliations for these systems, we find initial conditions for which the average change of the precession angle is zero or can be estimated asymptotically. Some more difficult cases are studied numerically.

math.CA↗
arXiv2018

On motions without falling of an inverted pendulum with dry friction

An inverted planar pendulum with horizontally moving pivot point is considered. It is assumed that the law of motion of the pivot point is given and the pendulum is moving in the presence of dry friction. Sufficient conditions for the existence of solutions along which the pendulum never falls below the horizontal positions are presented. The proof is based on the fact that solutions of the corresponding differential inclusion are right-unique and continuously depend on initial conditions, which is also shown in the paper.

math.CA↗
arXiv2017

On topological obstructions to global stabilization of an inverted pendulum

We consider a classical problem of control of an inverted pendulum by means of a horizontal motion of its pivot point. We suppose that the control law can be non-autonomous and non-periodic w.r.t. the position of the pendulum. It is shown that global stabilization of the vertical upward position of the pendulum cannot be obtained for any Lipschitz control law, provided some natural assumptions. Moreover, we show that there always exists a solution separated from the vertical position and along which the pendulum never becomes horizontal. Hence, we also prove that global stabilization cannot be obtained in the system where the pendulum can impact the horizontal plane (for any mechanical model of impact). Similar results are presented for several analogous systems: a pendulum on a cart, a spherical pendulum, and a pendulum with an additional torque control.

math.OC↗
arXiv2015

On forced oscillations in groups of interacting nonlinear systems

Consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them. Each subsystem is assumed to be a massive point moving with friction on a compact surface, possibly with a boundary, in an external periodic field. We present sufficient conditions for the existence of a periodic solution for the whole system. The result is illustrated by a series of examples including a chain of strongly coupled pendulums in a periodic field.

math.DS↗
arXiv2015

An inverted pendulum with a moving pivot point: examples of topological approach

Two examples concerning an application of topology in the study of the dynamics of an inverted plain mathematical pendulum with a pivot point moving along a horizontal straight line are considered. The first example is an application of the Wa{ż}ewski principle to the problem of the existence of a solution without falling in the case of a arbitrary prescribed law of motion of the pivot point. The second example is a proof of the existence of periodic solution in the same system when the law of motion is periodic as well. Moreover, in the second case it is also shown that along the obtained periodic solution the pendulum never becomes horizontal (falls). The proof is an example of application of the recent developments in the fixed point theory based on the Lefschetz-Hopf theory.

math.DS↗
arXiv2015

Periodic and falling-free motion of an inverted spherical pendulum with a moving pivot point

For the system of an inverted spherical pendulum with friction and a periodically moving pivot point we prove the existence of at least one periodic solution with the additional property of being falling-free. The last means that the pendulum never becomes horizontal along the considered periodic solution. Presented proof is an application of some recent results in the fixed point theory.

math.DS↗
arXiv2015

On forced oscillations of a constrained relativistic particle

We consider the forced motion of a relativistic particle constrained on a curve and present sufficient conditions for periodic oscillations by means of an illustrative geometrical approach. Obtained result is illustrated by a few examples including the forced relativistic pendulum.

math.DS↗