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O. Makarenkov

Publications and source records attributed to O. Makarenkov.

5 recordsLinked to original sources
arXiv2016

A new test for stick-slip limit cycles in dry-friction oscillators with small nonlinear friction characteristics

We consider a dry friction oscillator on a moving belt with both the Coulomb friction and a small nonlinear addition which can model e.g. the Stribeck effect. By using the perturbation theory, we establish a new sufficient condition for the nonlinearity to ensure the occurrence of a stick-slip limit cycle. The test obtained is more accurate compared to what one gets by building upon the divergence test.

math.DS↗
arXiv2009

Bifurcations from nondegenerate families of periodic solutions

By a nondegenerate $k$-parameterized family $K$ of periodic solutions we understand the situation when the geometric multiplicity of the multiplier +1 of the linearized on $K$ system equals to $k.$ Bifurcation of asymptotically stable periodic solutions from $K$ is well studied in the literature and different conditions have been proposeddepending on whether the algebraic multiplicity of +1 is $k$ or not (by Malkin, Loud, Melnikov, Yagasaki). In this paper we assume that the later is unknown. Asymptotic stability can not be understood in this case, but we demonstrate that the information about uniqueness of periodic solutions is still available. Moreover, we show that differentiability of the right hand sides is not necessary for the results of this kind and our theorems are proven under a kind of Lipschitz continuity.

math.CA↗
arXiv2009

Asymptotic stability of forced oscillations emanating from a limit cycle

Classical conditions for asymptotic stability of periodic solutions bifurcating from a limit cycle rely on the derivative of the corresponding bifurcation function F at the bifurcation point t. We show that for analytic systems this result is the one of topological nature, namely the topological index ind(t,F) is conclusive versus the sign of the derivative (d/dt)F(t). This allows to detect asymptotic stability also in the case when (d/dt)F(t)=0.

math.CA↗
arXiv2007

Synchronization problems for unidirectional feedback coupled nonlinear systems

In this paper we consider three different synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented.

math.CA↗