Bifurcation without Parameters

Liebscher, Stef An

doi:10.1007/978-3-319-10777-6

Cited by 34 publications

(33 citation statements)

References 44 publications

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“…The previous remarks drive the stability analysis of equilibria in memristive circuits to the mathematical context considered in [4,14,15,16,25]. In this setting, the existence of an m-dimensional manifold of equilibria implies that at least m eigenvalues of the linearization of the vector field at any of these equilibria are null.…”

Section: Tbwp In Memristive Circuitsmentioning

confidence: 99%

“…This problem must be framed in the theory of bifurcation without parameters originally introduced in the seminal papers [14,15,16]; cf. also the recent book [25]. When normal hyperbolicity fails, a change in the local qualitative properties typically occurs along the equilibrium manifold, hence the "bifurcation without parameters" term.…”

Section: Introductionmentioning

confidence: 98%

“…This is actually a three-fold goal. First, the characterization of the TBWP in [14] (and also in [25]) is only addressed for two-dimensional dynamics. However, most nonlinear circuits involve a large number of dynamic variables and a two-dimensional model reduction is rarely feasible.…”

Section: Introductionmentioning

confidence: 99%

“…However, most nonlinear circuits involve a large number of dynamic variables and a two-dimensional model reduction is rarely feasible. For their results to apply to higher dimensional problems, the authors assume in [14,25] that a prior reduction to a center two-dimensional manifold has been performed, but no explicit conditions paving the way for an appropriate reduction are given in arbitrary (finite) dimension. As a somewhat natural (yet not trivial) extension of their characterization we will present a TBWP theorem for explicit ODEs in R n , addressing the geometrical conditions which allow for a center manifold reduction in which the two-dimensional conditions of [14] do hold.…”

Section: Introductionmentioning

confidence: 99%

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Transcritical Bifurcation without Parameters in Memristive Circuits

Riaza¹

2018

SIAM J. Appl. Math.

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The transcritical bifurcation without parameters (TBWP) describes a stability change along a line of equilibria, resulting from the loss of normal hyperbolicity at a given point of such a line. Memristive circuits systematically yield manifolds of non-isolated equilibria, and in this paper we address a systematic characterization of the TBWP in circuits with a single memristor. To achieve this we develop two mathematical results of independent interest; the first one is an extension of the TBWP theorem to explicit ordinary differential equations (ODEs) in arbitrary dimension; the second result drives the characterization of this phenomenon to semiexplicit differential-algebraic equations (DAEs), which provide the appropriate framework for the analysis of circuit dynamics. In the circuit context the analysis is performed in graph-theoretic terms: in this setting, our first working scenario is restricted to passive problems (exception made of the bifurcating memristor), and in a second step some results are presented for the analysis of non-passive cases. The latter context is illustrated by means of a memristive neural network model. Ministerio de Economía y Competitividad (MINECO)/Fondo Europeo de Desarrollo Regional (FEDER).1. Let P be a matrix that drives f ′ (0) to Jordan form:2. Under the change of coordinates x = P y, the system x ′ = f (x) is transformed into y ′ =f (y) = P −1 f (P y), which reads as1. The circuit displays neither VMC-loops nor ILC-cutsets.2. There is a unique VML-loop, which includes the memristor and at least one inductor.

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Section: Tbwp In Memristive Circuitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transcritical Bifurcation without Parameters in Memristive Circuits

Riaza¹

2018

SIAM J. Appl. Math.

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“…If λ max (J T (p S * )) < 0, then every transverse eigenvalue of J M (x) has negative real part. By the Shoshitaishvili Reduction Principle, there exists a neighborhood V ∈ R 2N of x that is positively invariantly foliated by a family of stable manifolds corresponding to the family of stationary solutions in U (see [38]- [40]), each stable manifold spanned by the (generalized) eigenvectors associated with the N negative transverse eigenvalues of Proof. This follows from Definition V.1 and Lemma 1.…”

Section: B Stability Of Infection-free Equilibriamentioning

confidence: 99%

Adaptive Susceptibility and Heterogeneity in Contagion Models on Networks

Pagliara

Leonard

2021

IEEE Trans. Automat. Contr.

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Contagious processes, such as spread of infectious diseases, social behaviors, or computer viruses, affect biological, social, and technological systems. Epidemic models for large populations and finite populations on networks have been used to understand and control both transient and steady-state behaviors. Typically it is assumed that after recovery from an infection, every agent will either return to its original susceptible state or acquire full immunity to reinfection. We study the network SIRI (Susceptible-Infected-Recovered-Infected) model, an epidemic model for the spread of contagious processes on a network of heterogeneous agents that can adapt their susceptibility to reinfection. The model generalizes existing models to accommodate realistic conditions in which agents acquire partial or compromised immunity after first exposure to an infection. We prove necessary and sufficient conditions on model parameters and network structure that distinguish four dynamic regimes: infection-free, epidemic, endemic, and bistable. For the bistable regime, which is not accounted for in traditional models, we show how there can be a rapid resurgent epidemic after what looks like convergence to an infection-free population. We use the model and its predictive capability to show how control strategies can be designed to mitigate problematic contagious behaviors.

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Chaotic Dynamics in Spinning Shafts with Non-constant Rotating Speed Described by Variant Lyapunov Exponents

Georgiades

2020

Nonlinear Dynamics of Structures, Systems and Devices

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The dynamics of spinning shafts with non-constant rotating speed is described by a nonlinear system that under certain conditions might exhibit also chaotic behavior. In this article chaotic dynamics of the spinning shaft is examined. Initially, the trajectories in phase space around the equilibrium manifolds are determined. Then by choosing a set of initial conditions, nearby to an equilibrium, corresponding to eigenvalues of the Jacobian with a nonzero real part, identification of chaos is examined. Approximations of the trajectory, with the linearization curves around the equilibria, are defined and they are good in a region very close to the associated equilibrium point. It is shown that the eigenvalues, as Lyapunov exponents indicators, are not parameter dependent but state dependent. The eigenvalues of the linearized system within an orbit are varying from positive to zero, therefore the Lyapunov exponent is not defined through this limit as an explicit number but variant. The existence of eigenvalues with positive real parts in certain parts of the orbit is an indication of chaos since it shows a divergence of nearby orbits. One orbit starting from an initial condition which corresponds to eigenvalues with positive real part is crossing the threshold and pass to points that the eigenvalues with zero real parts, therefore this 'threshold' is not discriminating chaotic with regular regions as expected. The variant positive Lyapunov exponents have been examined also with numerical investigations and it is an indication of chaos. The Poincare section indicates irregular motion and the approximated Information Entropy is relatively high, and both are indicating chaos. It should be highlighted that this is a mechanical system with variant real parts of eigenvalues as Lyapunov exponents within one orbit and the threshold is insufficient to distinguish chaotic from regular regions. Further work is needed to determine the chaotic regions of the spinning shaft. Further developments in the mathematics of nonlinear dynamical systems associated with the equilibrium manifolds are needed to examine the significance of variant Lyapunov exponents for this kind of systems. Also, the necessity to reexamine the validity of existing algorithms and the development of new ones for the determination of variant Lyapunov exponents, become evident.

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Bifurcation without Parameters

Cited by 34 publications

References 44 publications

Transcritical Bifurcation without Parameters in Memristive Circuits

Transcritical Bifurcation without Parameters in Memristive Circuits

Adaptive Susceptibility and Heterogeneity in Contagion Models on Networks

Chaotic Dynamics in Spinning Shafts with Non-constant Rotating Speed Described by Variant Lyapunov Exponents

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