Over the last decade it has become evident that continuous-time (CT) dynamic circuits containing memristors are capable of exhibiting the coexistence of an “extreme” multitude of different attractors. This distinctive property stems from the so-called foliation feature, i.e., the state space of the circuit is composed of infinitely many invariant manifolds to which the dynamics is constrained. More recently, it has been shown that this feature is preserved by the discrete-time (DT) maps derived by discretizing the CT circuit using suitable procedures. In this paper we focus on a CT memristor-capacitor circuit known to possess only equilibrium points as attractors. Specifically, the circuit is a simple memristive cell comprising the parallel interconnection of a capacitor, a passive ideal flux-controlled memristor, and an active resistor. First, the differential equation governing the CT cell is discretized through a foliation-preserving procedure based on a convex combination of Forward and Backward Euler methods. This leads to a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(T,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dependent DT map (where T and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> are the discretization time step and the mixing parameter, respectively) that preserves the foliation feature of the CT circuit for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and any <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda \in [0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. Then, to illustrate the capacity to generate very complex dynamics, we show that these DT maps can exactly reproduce the dynamics of several well-known first-order and second-order chaotic maps (e.g., Circle, Logistic, Hénon and Lozi Maps) on their invariant manifolds, provided that specific embedding conditions involving T, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>, the memristor nonlinear characteristic, and the manifold index are satisfied. Furthermore, under mild assumptions on the parameters of the embedded chaotic map, the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(T,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-dependent DT map enjoys an intrinsic robustness property that leads to “extreme” multistability phenomena, i.e., the coexistence of infinitely many similar convergent, oscillatory and chaotic behaviors on nearby invariant manifolds. Finally, the Logistic Map is used as a case study to illustrate the DT map derived by solving the embedding conditions, and to discuss robustness of its dynamic behaviors.

Versatile discrete-time memristive cell

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2026年5月12日

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Giacomo Innocenti,
Alberto Tesi,
Mauro Di Marco,
Luca Pancioni,
Mauro Forti

摘要

Over the last decade it has become evident that continuous-time (CT) dynamic circuits containing memristors are capable of exhibiting the coexistence of an “extreme” multitude of different attractors. This distinctive property stems from the so-called foliation feature, i.e., the state space of the circuit is composed of infinitely many invariant manifolds to which the dynamics is constrained. More recently, it has been shown that this feature is preserved by the discrete-time (DT) maps derived by discretizing the CT circuit using suitable procedures. In this paper we focus on a CT memristor-capacitor circuit known to possess only equilibrium points as attractors. Specifically, the circuit is a simple memristive cell comprising the parallel interconnection of a capacitor, a passive ideal flux-controlled memristor, and an active resistor. First, the differential equation governing the CT cell is discretized through a foliation-preserving procedure based on a convex combination of Forward and Backward Euler methods. This leads to a $(T,\lambda )$ $(T, λ)$ -dependent DT map (where T and $\lambda $ $λ$ are the discretization time step and the mixing parameter, respectively) that preserves the foliation feature of the CT circuit for any $T>0$ $T > 0$ and any $\lambda \in [0,1]$ $λ \in [0, 1]$ . Then, to illustrate the capacity to generate very complex dynamics, we show that these DT maps can exactly reproduce the dynamics of several well-known first-order and second-order chaotic maps (e.g., Circle, Logistic, Hénon and Lozi Maps) on their invariant manifolds, provided that specific embedding conditions involving T, $\lambda $ $λ$ , the memristor nonlinear characteristic, and the manifold index are satisfied. Furthermore, under mild assumptions on the parameters of the embedded chaotic map, the $(T,\lambda )$ $(T, λ)$ -dependent DT map enjoys an intrinsic robustness property that leads to “extreme” multistability phenomena, i.e., the coexistence of infinitely many similar convergent, oscillatory and chaotic behaviors on nearby invariant manifolds. Finally, the Logistic Map is used as a case study to illustrate the DT map derived by solving the embedding conditions, and to discuss robustness of its dynamic behaviors.