On Discontinuous Piecewise Linear Models for Memristor Oscillators

Amador, Andrés; Freire, Emilio; Ponce, Enrique; Ros, Javier

doi:10.1142/s0218127417300221

Cited by 16 publications

(14 citation statements)

References 19 publications

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“…1. The frequency ω 0 is assumed to be a given positive constant due to the periodic input, while the offset A, the amplitude B, B > 0, and the phase difference Φ 0 , Φ 0 ∈ [0, 2π), between y 0 (τ ) and the harmonic forcing term in (25) are unknown.…”

Section: Computation Of Ppss Via the Hbmmentioning

confidence: 99%

“…The HBM consists in first substituting the approximation ( 26) into (25), then balancing the continuous and the first harmonic terms, and finally solving for A, B and Φ 0 the resulting system of equations. To proceed,…”

Section: Computation Of Ppss Via the Hbmmentioning

confidence: 99%

“…The peculiar mathematical property of (ideal) memristor circuits is that their state space is foliated, i.e., it amounts to a continuum of invariant manifolds where the circuit exhibits different lower-order dynamics. As an example, it is shown in [25,26] that the state space of a third-order oscillatory circuit with a single ideal memristor can be foliated in two-dimensional invariant manifolds and that the circuit dynamics is topologically equivalent to the reduced second-order dynamics on each manifolds, although the latter dynamics depends on an extra parameter that is function of the initial conditions. The reduced dynamics is also described by a smoother vector field.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Prediction of period doubling bifurcations in harmonically forced memristor circuits

et al. 2019

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The paper studies bifurcations and complex dynamics in a class of nonautonomous oscillatory circuits with a flux-controlled memristor and harmonic forcing term. It is first shown that, as in the autonomous case, the state space of any memristor circuit of the class can be decomposed in invariant manifolds. It turns out that the memristor circuit dynamics is given by the collection of the dynamics of a family of circuits, with a nonlinear resistor in place of the memristor, which is parameterized by an additional constant input whose value depends on the initial conditions of the memristor circuit. This property makes it possible to employ the Harmonic Balance Method in order to study the periodic solutions and their bifurcations due to changing the amplitude and the frequency of the harmonic input on a fixed manifold or due to changing the initial conditions for a fixed harmonic input. The main result is that in both these cases the Harmonic Balance Method is quite effective to accurately predict period-doubling bifurcations of the periodic solutions. Analytical predictions are obtained in the cases of linear-plus-cubic and piece-wise linear memristor flux-charge characteristics.

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Section: Computation Of Ppss Via the Hbmmentioning

confidence: 99%

Section: Computation Of Ppss Via the Hbmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Prediction of period doubling bifurcations in harmonically forced memristor circuits

et al. 2019

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“…Later, it became clear that the state space of circuits containing ideal memristors can be decomposed into a continuum of invariant manifolds (foliation property of the state space), which are indexed by some constant parameter whose value depends on the initial conditions of the circuit. Specifically, in [22,23] a third-order memristor circuit is investigated in the voltage-current domain, while quite general classes of memristor circuits are analyzed in the flux-charge domain (see [24][25][26][27] and references therein). In particular, the flux-charge analysis method (FCAM), introduced in [24,25], makes it clear that the rich dynamics displayed by memristor circuits is due to the fact that the state space contains infinitely many invariant manifolds.…”

Section: Introductionmentioning

confidence: 99%

Circuits with a mem-element: invariant manifolds control via pulse programmed sources

et al. 2021

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The paper considers the problem of controlling multistability in a general class of circuits composed of a linear time-invariant two-terminal (one port) element, containing linear R, L, C components and ideal operational amplifiers, coupled with one of the mem-elements (memory elements) introduced by Prof. L.O. Chua, i.e., memristors, memcapacitors, and meminductors. First, explicit expressions of the invariant manifolds of the circuit are directly given in terms of the state variables of the two-terminal element and the mem-element. Then, the problem of steering the circuit dynamics from an initial invariant manifold to a final one, and hence to potentially switch among different attractors of the circuit, is addressed by designing pulse shaped control inputs. The control inputs ensure that the transition between the initial and final manifolds is accomplished within a given finite time interval. Moreover, it is shown how the designed control inputs can be implemented by introducing independent voltage and current sources in the two-terminal element. Notably, it turns out that it is always possible to solve the considered control problem by using a unique independent source. Several examples are provided to illustrate how the proposed approach can be applied to different circuits with mem-elements and to highlight the influence of the features of the designed sources on the behavior of the controlled dynamics.

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“…it can be decomposed in a continuum of invariant manifolds where the circuit dynamics is described by a reduced order system. Specifically, in [Amador et al, 2017;Ponce et al, 2017] it is shown that the dynamics of a third order memristor circuit admits a first integral in the current-voltage domain and hence it can be equivalently described by a family of second order systems indexed by an additional constant parameter. Notably, the existence of a first integral implies that the second order systems have a smoother vector field, which is a useful property when the memristor has a piecewise linear characteristic.…”

Section: Introductionmentioning

confidence: 99%

Input–Output Characterization of the Dynamical Properties of Circuits with a Memelement

Innocenti

Marco

Tesi

et al. 2020

Int. J. Bifurcation Chaos

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The paper proposes a novel input–output approach to characterize the dynamical properties of a class of circuits composed by a linear time-invariant two-terminal element coupled with one of the ideal memelements (memory elements) introduced by Prof. L. O. Chua, i.e. memristors, memcapacitors, and meminductors. The developed approach permits to readily determine the conditions under which the dynamics of any circuit of the class admits a first integral. It is also shown that the circuit dynamics can be obtained by collecting the dynamical behaviors displayed by a canonical reduced-order input–output system subject to a constant input of any amplitude. Notably, the reduced-order system exactly describes the dynamics of a circuit forced by a constant generator and with a nonlinear memoryless element in place of the memelement. The relation between the proposed input–output approach and the available state space ones (e.g. Flux-Charge Analysis Method (FCAM)) is also addressed. The main result is that explicit expressions of the invariant manifolds can be directly obtained in the voltage–current state space. Finally, it is shown how the approach also applies to circuits which contain forcing generators. It is believed that the proposed input–output approach can be a useful alternative to state space methods for studying multistability and control issues.

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On Discontinuous Piecewise Linear Models for Memristor Oscillators

Cited by 16 publications

References 19 publications

Prediction of period doubling bifurcations in harmonically forced memristor circuits

Prediction of period doubling bifurcations in harmonically forced memristor circuits

Circuits with a mem-element: invariant manifolds control via pulse programmed sources

Input–Output Characterization of the Dynamical Properties of Circuits with a Memelement

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