Effect of positive feedback on Devil's staircase input-output relationship
Alexei Pokrovskii Dmitrii Rachinskii
Discrete & Continuous Dynamical Systems - S 2013, 6(4): 1095-1112 doi: 10.3934/dcdss.2013.6.1095
We consider emerging hysteresis behaviour in a closed loop systemthat includes a nonlinear link $f$ of the Devil's staircase (Cantorfunction) type and a positive feedback. This type of closed loopsarises naturally in analysis of networks where local ``negative''coupling of network elements is combined with ``positive'' couplingat the level of the mean-field interaction (in the limit case whenthe impact of each individual vertex is infinitesimal, while thenumber of vertices is growing). For the Cantor function $f$, takenas a model, and for a monotonically increasing input, we present thecorresponding output of the system explicitly, showing that theoutput is piecewise constant and has a finite number of equal jumps.We then discuss hysteresis loops of the system for genericnon-monotone inputs. The results are presented in the context of differential equations describingnonlinear control systems with almost immediate linear feedback, i.e., in the limit where the time of propagation of the signalthrough the feedback loop tends to zero.
keywords: Transducer feedback differential system delay Cantor function hysteresis loop non-ideal relay Preisach operator.
Asymptotics of the Arnold tongues in problems at infinity
Victor Kozyakin Alexander M. Krasnosel’skii Dmitrii Rachinskii
Discrete & Continuous Dynamical Systems - A 2008, 20(4): 989-1011 doi: 10.3934/dcds.2008.20.989
We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$, and study their trajectories of large amplitudes. The expansion of the map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller vanishing part. We study Arnold tongues: the sets of parameter values for which the large-amplitude periodic trajectories exist. The Arnold tongues in problems at infinity generically are thick triangles [4]; here we obtain asymptotic estimates for the length of the Arnold tongues and for the length of their triangular part. These estimates allow us to study subfurcation at infinity. In the related problems on small-amplitude periodic orbits near an equilibrium, similarly defined Arnold tongues have the form of narrow beaks. For standard pictures associated with the Neimark-Sacker bifurcation of smooth discrete time systems at an equilibrium, the Arnold tongues have asymptotically zero width except for the strong resonance points. The different shape of the tongues in the problem at infinity is due to the non-polynomial form of the principal homogeneous nonlinear term of the map $U(\cdot;\lambda)$: this form implies non-degeneracy of the nonlinear terms in the expansion of the map iterations and non-degeneracy of the corresponding resonance functions.
keywords: periodic trajectory discrete time system Bifurcation at infinity Poincare map Arnold tongue subfurcation positively homogeneous nonlinearity saturation invariant set rotation of vector fields. subharmonics
Restrictions to the use of time-delayed feedback control in symmetric settings
Edward Hooton Pavel Kravetc Dmitrii Rachinskii
Discrete & Continuous Dynamical Systems - B 2018, 23(2): 543-556 doi: 10.3934/dcdsb.2017207

We consider the problem of stabilization of unstable periodic solutions to autonomous systems by the non-invasive delayed feedback control known as Pyragas control method. The Odd Number Theorem imposes an important restriction upon the choice of the gain matrix by stating a necessary condition for stabilization. In this paper, the Odd Number Theorem is extended to equivariant systems. We assume that both the uncontrolled and controlled systems respect a group of symmetries. Two types of results are discussed. First, we consider rotationally symmetric systems for which the control stabilizes the whole orbit of relative periodic solutions that form an invariant two-dimensional torus in the phase space. Second, we consider a modification of the Pyragas control method that has been recently proposed for systems with a finite symmetry group. This control acts non-invasively on one selected periodic solution from the orbit and targets to stabilize this particular solution. Variants of the Odd Number Limitation Theorem are proposed for both above types of systems. The results are illustrated with examples that have been previously studied in the literature on Pyragas control including a system of two symmetrically coupled Stewart-Landau oscillators and a system of two coupled lasers.

keywords: Stabilization of periodic orbits Pyragas control delayed feedback S1-equivariance finite symmetry group
Arnold tongues for bifurcation from infinity
Victor S. Kozyakin Alexander M. Krasnosel’skii Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - S 2008, 1(1): 107-116 doi: 10.3934/dcdss.2008.1.107
We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$. The map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller part. We describe the sets of parameter values for which the large-amplitude $n$-periodic trajectories exist for a fixed $n$. In the related problems on small periodic orbits near zero, similarly defined parameter sets, known as Arnold tongues, are more narrow.
keywords: Arnold tongue positively homogeneous nonlinearity discrete time system bifurcation at infinity Poincare map. Periodic trajectory saturation
Linear stability analysis of systems with Preisach memory
Alexander Pimenov Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - B 2009, 11(4): 997-1018 doi: 10.3934/dcdsb.2009.11.997
We consider differential equations coupled with the input-output memory relation defined by the Preisach operator. The differential equation relates an instant value of the rate of change of the output of the Preisach operator with an instant value of its input. We propose an algorithm for the linearisation of the evolution operator of the system and apply it to define the characteristic multiplier of periodic solutions, which determines their stability. Examples of the system considered include models of terrestrial hydrology and electronic oscillators with hysteresis.
keywords: local stability analysis operator-differential equation Preisach model. Periodic solution
Peter E. Kloeden Alexander M. Krasnosel'skii Pavel Krejčí Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - B 2013, 18(2): i-iii doi: 10.3934/dcdsb.2013.18.2i
Alexei Vadimovich Pokrovskii was an outstanding mathematician, a scientist with very broad mathematical interests, and a pioneer in the mathematical theory of systems with hysteresis. He died unexpectedly on September 1, 2010 at the age 62. For the previous nine years he had been Professor and Head of Applied Mathematics at University College Cork in Ireland.

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Kurzweil integral representation of interacting Prandtl-Ishlinskii operators
Pavel Krejčí Harbir Lamba Sergey Melnik Dmitrii Rachinskii
Discrete & Continuous Dynamical Systems - B 2015, 20(9): 2949-2965 doi: 10.3934/dcdsb.2015.20.2949
We consider a system of operator equations involving play and Prandtl-Ishlinskii hysteresis operators. This system generalizes the classical mechanical models of elastoplasticity, friction and fatigue by introducing coupling between the operators. We show that under quite general assumptions the coupled system is equivalent to one effective Prandtl-Ishlinskii operator or, more precisely, to a discontinuous extension of the Prandtl-Ishlinskii operator based on the Kurzweil integral of the derivative of the state function. This effective operator is described constructively in terms of the parameters of the coupled system. Our result is based on a substitution formula which we prove for the Kurzweil integral of regulated functions integrated with respect to functions of bounded variation. This formula allows us to prove the composition rule for the generalized (discontinuous) Prandtl-Ishlinskii operators. The composition rule, which underpins the analysis of the coupled model, then establishes that a composition of generalized Prandtl-Ishlinskii operators is also a generalized Prandtl-Ishlinskii operator provided that a monotonicity condition is satisfied.
keywords: hysteresis operator network model. composition formula Kurzweil integral discontinuous Prandtl-Ishlinskii operator Regulated function substitution formula
Realization of arbitrary hysteresis by a low-dimensional gradient flow
Dmitrii Rachinskii
Discrete & Continuous Dynamical Systems - B 2016, 21(1): 227-243 doi: 10.3934/dcdsb.2016.21.227
We consider gradient systems with an increasing potential that depends on a scalar parameter. As the parameter is varied, critical points of the potential can be eliminated or created through saddle-node bifurcations causing the system to transit from one stable equilibrium located at a (local) minimum point of the potential to another minimum along the heteroclinic connections. These transitions can be represented by a graph. We show that any admissible graph has a realization in the class of two dimensional gradient flows. The relevance of this result is discussed in the context of genesis of hysteresis phenomena. The Preisach hysteresis model is considered as an example.
keywords: gradient system hysteresis Multi-stability graph. heteroclinic connection
Periodic canard trajectories with multiple segments following the unstable part of critical manifold
Alexander M. Krasnosel'skii Edward O'Grady Alexei Pokrovskii Dmitrii I. Rachinskii
Discrete & Continuous Dynamical Systems - B 2013, 18(2): 467-482 doi: 10.3934/dcdsb.2013.18.467
We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on $n$ scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes $n$ fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has $n$ portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.
keywords: stability. canard trajectory degree theory Periodic solution
Sliding Hopf bifurcation in interval systems
Hooton Edward Balanov Zalman Krawcewicz Wieslaw Rachinskii Dmitrii
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 3545-3566 doi: 10.3934/dcds.2017152

Abstract. In this paper, the equivariant degree theory is used to analyze the occurrence of the Hopf bifurcation under effectively verifiable mild conditions. We combine the abstract result with standard interval polynomial techniques based on Kharitonov's theorem to show the existence of a branch of periodic solutions emanating from the equilibrium in the settings relevant to robust control. The results are illustrated with a number of examples.

keywords: Hopf bifurcation interval polynomial non-local branch Kharitonov's theorem zero exclusion principle S1-degree

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