Equivariant Global Hopf Bifurcation in Abstract Nonlinear Parabolic Equations
In this paper we study local and global symmetric Hopf bifurcation in abstract parabolic systems by means of the twisted equivariant degree.
math.DS↗
arXiv subjects
Dmitrii Rachinskii
Publications and source records attributed to Dmitrii Rachinskii.
Hysteresis resulting from Lennard-Jones interactions
The fundamental mechanism of hysteresis in the quasistatic limit of multi-stable systems is associated with transitions of the system from one local minimum of the potential energy to another. In this scenario, as system parameters are (quasistatically) varied, the transition is prompted when a saddle-node bifurcation eliminates the minimum where the system resides in. The objective of the present work is to specify this generic mechanism for systems of interacting particles assuming a natural single-well (Lennard-Jones) interaction potential for each pair of particles. We show multi-stability and present details of hysteresis scenarios with the associated bifurcations and transitions in a case study of constrained four-degrees-of-freedom four particle systems on the plane.
cond-mat.mtrl-sci↗
Dynamics of inertial pair coupled via frictional interface
Understanding the dynamics of two inertial bodies coupled via a friction interface is essential for a wide range of systems and motion control applications. Coupling terms within the dynamics of an inertial pair connected via a passive frictional contact are non-trivial and have long remained understudied in system communities. This problem is particularly challenging from a point of view of modeling the interaction forces and motion state variables. This paper deals with a generalized motion problem in systems with a free (of additional constraints) friction interface, assuming the classical Coulomb friction with discontinuity at the velocity zero crossing. We formulate the dynamics of motion as the closed-form ordinary differential equations containing the sign operator for mapping both, the Coulomb friction and the switching conditions, and discuss the validity of the model in the generalized force and motion coordinates. The system has one active degree of freedom (the driving body) and one passive degree of freedom (the driven body). We demonstrate the global convergence of trajectories for a free system with no external excitation forces. Then, an illustrative case study is presented for a harmonic oscillator with a frictionally coupled second mass that is not grounded or connected to a fixed frame. This simplified example illustrates a realization and main features of the proposed (general) modeling framework. Some future development and related challenges are discussed at the end of the paper.
eess.SY↗
Global stability of SIR model with heterogeneous transmission rate modeled by the Preisach operator
In recent years, classical epidemic models, which assume stationary behavior of individuals, have been extended to include an adaptive heterogeneous response of the population to the current state of the epidemic. However, it is widely accepted that human behavior can exhibit history-dependence as a consequence of learned experiences. This history-dependence is similar to hysteresis effects that have been well-studied in control theory. To illustrate the importance of history-dependence for epidemic theory, we study dynamics of a variant of the SIRS model where individuals exhibit lazy-switch responses to prevalence dynamics. The resulting model, which includes the Preisach hysteresis operator, possesses a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected and recovered populations. We discuss stability properties of the endemic equilibrium set and relate them to the degree of heterogeneity of the adaptive response. Our results support the argument that public health responses during the emergence of a new disease can have long-term consequences for subsequent management efforts. The main mathematical contribution of this work is a method of global stability analysis, which uses a family of Lyapunov functions corresponding to different branches of the hysteresis operator.
math.DS↗
Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps
We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear~naturally.
math.DS↗
Dynamics of SIR model with heterogeneous response to intervention policy
We study dynamics of a variant of the SIR the model, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. The degree of heterogeneity of the response is measured by the variance $σ^2$ of the corresponding distribution (the Preisach density function). The proposed model has a continuum of endemic equilibrium states characterized by different proportions of susceptible, infected and recovered populations. We consider how the limit point of the epidemic trajectory and the infection peak along this trajectory depend on $σ$. The heterogeneous model is compared to the the switched model with an ideally uniform response corresponding to $σ=0$.
math.DS↗
Effect of diagnostic testing on the isolation rate in a compartmental model with asymptomatic groups
We present two epidemiological models, which extend the classical SEIR model by accounting for the effect of indiscriminate quarantining, isolation of infected individuals based on testing and the presence of asymptomatic individuals. Given a constraint that limits the maximal number of simultaneous active cases, we demonstrate that the isolation rate, which enforces this constraint, decreases with the increasing testing rate. The models predict that massive testing allows to control the infection spread using a much lower isolation rate than in the case of indiscriminate quarantining. We also show that dynamics of infection are more sensitive to the parameters of the asymptomatic groups than the corresponding parameters of the symptomatic groups.
math.DS↗
Mixed global dynamics of forced vibro-impact oscillator with Coulomb friction
The paper revisits a well-known model of forced vibro-impact oscillator with Amonton-Coulomb friction. In vast majority of the existing studies, this model included also viscous friction, and its global dynamics in the state space is governed by periodic, quasiperiodic or chaotic attractors. We demonstrate that removal of the viscous friction leads to qualitative modification of the global dynamics. Namely, the state space is divided into the regions with "regular" attraction to the aforementioned special solutions, and the regions with profoundly Hamiltonian dynamics. The latter regions contain structures typical for forced Hamiltonian systems: stability islands, extended non-attractive chaotic regions etc. We prove that such local Hamiltonian behavior should occur for phase trajectories with non-vanishing velocity. Stability analysis for the periodic orbits confirms the above statement. It is demonstrated that similar mixed global dynamics can be observed in broader class of models.
math.DS↗
A continuum of path-dependent equilibrium solutions induced by sticky expectations
We analyze a simple macroeconomic model where rational inflation expectations is replaced by a boundedly rational, and genuinely sticky, response to changes in the actual inflation rate. The stickiness is introduced in a novel way using a mathematical operator that is amenable to rigorous analysis. We prove that, when exogenous noise is absent from the system, the unique equilibrium of the rational expectations model is replaced by an entire line segment of possible equilibria with the one chosen depending, in a deterministic way, upon the previous states of the system. The agents are sufficiently far-removed from the rational expectations paradigm that problems o indeterminacy do not arise. The response to exogenous noise is far more subtle than in a unique equilibrium model. After sufficiently small shocks the system will indeed revert to the same equilibrium but larger ones will move the system to a different one (at the same model parameters). The path to this new equilibrium may be very long with a highly unpredictable endpoint. At certain model parameters exogenously-triggered runaway inflation can occur. Finally, we analyze a variant model in which the same form of sticky response is introduced into the interest rate rule instead.
math.DS↗
Global stability of a piecewise linear macroeconomic model with a continuum of equilibrium states and sticky expectation
We consider piecewise linear discrete time macroeconomic models, which possess a continuum of equilibrium states. These systems are obtained by replacing rational inflation expectations with a boundedly rational, and genuinely sticky, response of agents to changes in the actual inflation rate in a standard Dynamic Stochastic General Equilibrium model. Both for a low-dimensional variant of the model, with one representative agent, and the multi-agent model, we show that, when exogenous noise is absent from the system, the continuum of equilibrium states is the global attractor. Further, when a uniformly bounded noise is present, or the equilibrium states are destabilized by an imperfect Central Bank policy (or both), we estimate the size of the domain that attracts all the trajectories. The proofs are based on introducing a family of Lyapunov functions and, for the multi-agent model, deriving a formula for the inverse of the Prandtl-Ishlinskii operator acting in the space of discrete time inputs and outputs.
math.DS↗
The Devil is in the Details: Spectrum and Eigenvalue Distribution of the Discrete Preisach Memory Model
We consider the adjacency matrix associated with a graph that describes transitions between $2^{N}$ states of the discrete Preisach memory model. This matrix can also be associated with the last-in-first-out inventory management rule. We present an explicit solution for the spectrum by showing that the characteristic polynomial is the product of Chebyshev polynomials. The eigenvalue distribution (density of states) is explicitly calculated and is shown to approach a scaled Devil's staircase. The eigenvectors of the adjacency matrix are also expressed analytically.
math-ph↗
Chaos in Saw Map
We consider dynamics of a scalar piecewise linear "saw map" with infinitely many linear segments. In particular, such maps are generated as a Poincaré map of simple two-dimensional discrete time piecewise linear systems involving a saturation function. Alternatively, these systems can be viewed as a feedback loop with the so-called stop hysteresis operator. We analyze chaotic sets and attractors of the "saw map" depending on its parameters.
math.DS↗
Hopf Bifurcation of Relative Periodic Solutions: Case Study of a Ring of Passively Mode-Locked Lasers
In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting $Γ\times S^1$-spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted $Γ\times S^1$-degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group $Γ=D_8$.
math.DS↗
Equivariant Versions of Odd Number Theorem
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.
math.DS↗
Non-invasive stabilization of periodic orbits in $O_4$-symmetrically coupled Van der Pol oscillators
Pyragas time delayed feedback control has proven itself as an effective tool to non-invasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings and applied to stabilize branches of small periodic solutions in systems of symmetrically coupled Landau oscillators near a Hopf bifurcation point. The form of the control ensures the non-invasiveness property, hence reducing the problem to finding a set of the gain matrices, which would guarantee the stabilization. In this paper, we apply this method to a system of Van der Pol oscillators coupled in a cube-like configuration leading to $O_4$-equivariance. We discuss group theoretic restrictions which help to shape our choice of control. Furthermore, we explicitly describe the domains in the parameter space for which the periodic solutions are stable.
math.OC↗
Dynamics of Discrete Time Systems with a Hysteresis Stop Operator
We consider a piecewise linear two-dimensional dynamical system that couples a linear equation with the so-called stop operator. Global dynamics and bifurcations of this system are studied depending on two parameters. The system is motivated by modifications to general-equilibrium macroeconomic models that attempt to capture the frictions and memory-dependence of realistic economic agents.
math.DS↗
Pulsating dynamics of slow-fast population models with delay
We discuss a bifurcation scenario which creates periodic pulsating solutions in slow-fast delayed systems through a cascade of almost simultaneous Hopf bifurcations. This scenario has been previously associated with formation of pulses in a delayed model of mode-locked semiconductor lasers. In this work, through a case study of several examples, we establish that a cascade of Hopf bifurcations can produce periodic pulses, with a period close to the delay time, in population dynamics models and explore the conditions that ensure the realization of this scenario near a transcritical bifurcation threshold. We derive asymptotic approximations for the pulsating solution and consider scaling of the solution and its period with the small parameter that measures the ratio of the time scales. The role of competition for the realization of the bifurcation scenario is highlighted.
math.DS↗
Asymptotics of sign-changing patterns in hysteretic systems with diffusive thresholds
We consider a reaction-diffusion system including discontinuous hysteretic relay operators in reaction terms. This system is motivated by an epigenetic model that describes the evolution of a population of organisms which can switch their phenotype in response to changes of the state of the environment. The model exhibits formation of patterns in the space of distributions of the phenotypes over the range of admissible switching strategies. We propose asymptotic formulas for the pattern and the process of its formation.
math.AP↗