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.
On geometric conditions for reduction of the Moreau sweeping process to the Prandtl-Ishlinskii operator
Dmitrii Rachinskii
Discrete & Continuous Dynamical Systems - B 2018, 23(8): 3361-3386 doi: 10.3934/dcdsb.2018246

The sweeping process was proposed by J. J. Moreau as a general mathematical formalism for quasistatic processes in elastoplastic bodies. This formalism deals with connected Prandtl's elastic-ideal plastic springs, which can form a system with an arbitrarily complex topology. The model describes the complex relationship between stresses and elongations of the springs as a multi-dimensional differential inclusion (variational inequality). On the other hand, the Prandtl-Ishlinskii model assumes a very simple connection of springs. This model results in an input-output operator, which has many good mathematical properties and admits an explicit solution for an arbitrary input. It turns out that the sweeping processes can be reducible to the Prandtl-Ishlinskii operator even if the topology of the system of springs is complex. In this work, we analyze the conditions for such reducibility.

keywords: Differential inclusion elastoplasticity quasistatic process model reduction explicit solution hysteresis loop
Selective Pyragas control of Hamiltonian systems
Edward Hooton Pavel Kravetc Dmitrii Rachinskii Qingwen Hu
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 2019-2034 doi: 10.3934/dcdss.2019130

We consider a Newton system which has a branch (surface) of neutrally stable periodic orbits. We discuss sufficient conditions which allow arbitrarily small delayed Pyragas control to make one selected cycle asymptotically stable. In the case of small amplitude periodic solutions we give conditions in terms of the asymptotic expansion of the right hand side, while in the case of larger cycles we frame the conditions in terms of the Floquet modes of the target orbit as a solution of the uncontrolled system.

keywords: Time-delayed feedback stabilization of periodic solutions delay differential equations Hamiltonian systems
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
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

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