320
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
architectures that is well suited for implementation in nanotechnology [17–19].
The articles [20–23] explore the dynamics of arrays of diffusively coupled Chua’s
oscillators where the nonlinear resistor in each oscillator is replaced by a memristor.
By simulations and experiments, it is demonstrated that, due to the nonlinear
dynamics of memristors, such arrays are endowed with a new and intriguing variety
of complex spatiotemporal phenomena [24, 25]. Interestingly, several form of
synchronization can be observed by modifying not only circuit parameters and coupling strengths, but also initial conditions of dynamic elements and, especially, of
memristors. Considering that locally connected regular architectures are especially
well suited for nanoscale implementation, and that emerging dynamic phenomena
due to the presence of memristors are observed, it is crucial to develop analytical
and numerical tools to investigate complex dynamics including synchronization
phenomena in bio-inspired networks of memristor-based oscillatory cells.
This chapter considers a 1D array of N diffusively coupled memristor chaotic
circuits (MCCs), where each MCC is obtained by replacing the nonlinear resistor of
a Chua’s oscillator with a flux-controlled memristor (cf. Chap. 6). We name NMCC
the considered array of N interconnected MCCs. The goal is to analyze the complex
dynamics, bifurcations, and synchronization phenomena in NMCC by using FCAM.
The main contributions in this chapter can be summarized as follows:
• it is shown that the state space in the voltage-current (v, i)-domain can be foliated
in a continuum of manifolds that are invariant for the dynamics. Moreover, it is
possible to explicitly find the state equations (SEs) describing the reduced-order
dynamics on each invariant manifold. Compared to the result in Chap. 7, this
step is accomplished via a newly developed circuit technique based on writing
suitable sets of Kirchhoff laws for the NMCC in the (ϕ, q)-domain;
• via the concept of invariant manifolds, it is analytically shown that for the
NMCC there coexist infinitely many different complex attractors and dynamics,
and that bifurcations without parameters, i.e., bifurcations due to changes of
initial conditions for a fixed set of circuit parameters, occur. Such results give
an analytic explanation of initial-condition dependent nonlinear phenomena
experimentally observed and reported in several publications [20, 21, 24, 25];
• it is shown how the explicit knowledge of invariant manifolds, and the reducedorder dynamics on each manifold, enable to exploit results available in the
literature for analyzing some relevant features of the complex nonlinear dynamics
in the NMCC. In particular, the theoretic results in the chapter permit to choose
initial conditions such that the nonlinear dynamics in the NMCC take place on
a selected manifold with some special properties, named “zero-manifold,” see
Sect. 8.4 for details. By relying on previous analytic results in [26–28], various
types of synchronization phenomena on the “zero-manifold” are investigated,
including in-phase and/or anti-phase synchronization of periodic/chaotic attractors.
Although the chapter focuses on a 1D array of N diffusively coupled memristor
oscillators, all the obtained results can be extended, mutatis mutandis, to any
network of MCCs with diffusive couplings arranged in 2D or 3D structures.
8 Complex Dynamics and Synchronization Phenomena in Arrays of Memristor. . .
architectures that is well suited for implementation in nanotechnology [17–19].
The articles [20–23] explore the dynamics of arrays of diffusively coupled Chua’s
oscillators where the nonlinear resistor in each oscillator is replaced by a memristor.
By simulations and experiments, it is demonstrated that, due to the nonlinear
dynamics of memristors, such arrays are endowed with a new and intriguing variety
of complex spatiotemporal phenomena [24, 25]. Interestingly, several form of
synchronization can be observed by modifying not only circuit parameters and coupling strengths, but also initial conditions of dynamic elements and, especially, of
memristors. Considering that locally connected regular architectures are especially
well suited for nanoscale implementation, and that emerging dynamic phenomena
due to the presence of memristors are observed, it is crucial to develop analytical
and numerical tools to investigate complex dynamics including synchronization
phenomena in bio-inspired networks of memristor-based oscillatory cells.
This chapter considers a 1D array of N diffusively coupled memristor chaotic
circuits (MCCs), where each MCC is obtained by replacing the nonlinear resistor of
a Chua’s oscillator with a flux-controlled memristor (cf. Chap. 6). We name NMCC
the considered array of N interconnected MCCs. The goal is to analyze the complex
dynamics, bifurcations, and synchronization phenomena in NMCC by using FCAM.
The main contributions in this chapter can be summarized as follows:
• it is shown that the state space in the voltage-current (v, i)-domain can be foliated
in a continuum of manifolds that are invariant for the dynamics. Moreover, it is
possible to explicitly find the state equations (SEs) describing the reduced-order
dynamics on each invariant manifold. Compared to the result in Chap. 7, this
step is accomplished via a newly developed circuit technique based on writing
suitable sets of Kirchhoff laws for the NMCC in the (ϕ, q)-domain;
• via the concept of invariant manifolds, it is analytically shown that for the
NMCC there coexist infinitely many different complex attractors and dynamics,
and that bifurcations without parameters, i.e., bifurcations due to changes of
initial conditions for a fixed set of circuit parameters, occur. Such results give
an analytic explanation of initial-condition dependent nonlinear phenomena
experimentally observed and reported in several publications [20, 21, 24, 25];
• it is shown how the explicit knowledge of invariant manifolds, and the reducedorder dynamics on each manifold, enable to exploit results available in the
literature for analyzing some relevant features of the complex nonlinear dynamics
in the NMCC. In particular, the theoretic results in the chapter permit to choose
initial conditions such that the nonlinear dynamics in the NMCC take place on
a selected manifold with some special properties, named “zero-manifold,” see
Sect. 8.4 for details. By relying on previous analytic results in [26–28], various
types of synchronization phenomena on the “zero-manifold” are investigated,
including in-phase and/or anti-phase synchronization of periodic/chaotic attractors.
Although the chapter focuses on a 1D array of N diffusively coupled memristor
oscillators, all the obtained results can be extended, mutatis mutandis, to any
network of MCCs with diffusive couplings arranged in 2D or 3D structures.