6.4 Discussion
263
6.4 Discussion
Here, we collect some concluding remarks concerning the results obtained in the
chapter.
1. Several papers in the literature have studied the dynamics and bifurcations
of specific memristor circuits in the class LM. A summary of some main
contributions is reported next. The papers [11, 12] studied via simulations the
influence of initial conditions on the oscillatory behavior of a circuit in the class
LM pointing out the coexistence of several different attractors for the same set
of circuit parameters. A class of Chua’s oscillators containing a memristor is
studied in [18] and coexistence of different attractors, initial-condition dependent
bifurcations, and complex dynamics are highlighted by means of simulations.
In the article [19], a class of memristor cellular neural networks is considered
which is able to generate Turing patterns. The paper highlights by numerical
means how the initial conditions in the memristors influence the onset of Turing
patterns and also the types of Turing patterns generated by the network. The
paper [20] has experimentally studied the role of initial conditions in relation to
a type of bifurcations, named grazing bifurcations, for a fixed set of parameters,
in a memristor circuit with a nonsmooth memristor characteristic. Numerical
studies in the (v, i)-domain on the influence of initial conditions on dynamics and
bifurcations have been carried out also in [21–23]. The articles [24, 25] developed
a method in the (v, i)-domain to study bifurcations, and in particular bifurcations
without parameters, in cases where the DAE description of memristor circuits is
available but the SE description is not. However, it is not analytically investigated
how initial conditions are related to such bifurcations. Stability properties of
attractors, local and global bifurcations, and the role of the initial conditions have
been extensively investigated in [26] as well. We refer the reader to [27, 28], and
references therein, for other contributions along this line.
Overall, the quoted papers aim at highlighting peculiar phenomena observable
in memristor circuits as the coexistence of a huge number of different attractors
for a fixed set of circuit parameters [10]. Moreover, bifurcations due to changing
initial conditions, for a fixed set of circuit parameters, are highlighted. One
fundamental shortcoming of these contributions is that such phenomena are
investigated basically by numerical or experimental means, while an analytic
explanation is lacking.
In this chapter we have shown, by discussing some significant applications,
that FCAM is effective to analytically explain these initial-condition related
phenomena and is especially well suited to rigorously prove the coexistence
of infinitely many different attractors and the presence of bifurcations without
parameters in memristor circuits. Such an extremely rich and complex dynamic
scenario corresponds to a case referred to in the literature as extreme multistability. The reader is referred to [10] for other classes of nonlinear dynamical systems
in physics and engineering featuring extreme multistability.
263
6.4 Discussion
Here, we collect some concluding remarks concerning the results obtained in the
chapter.
1. Several papers in the literature have studied the dynamics and bifurcations
of specific memristor circuits in the class LM. A summary of some main
contributions is reported next. The papers [11, 12] studied via simulations the
influence of initial conditions on the oscillatory behavior of a circuit in the class
LM pointing out the coexistence of several different attractors for the same set
of circuit parameters. A class of Chua’s oscillators containing a memristor is
studied in [18] and coexistence of different attractors, initial-condition dependent
bifurcations, and complex dynamics are highlighted by means of simulations.
In the article [19], a class of memristor cellular neural networks is considered
which is able to generate Turing patterns. The paper highlights by numerical
means how the initial conditions in the memristors influence the onset of Turing
patterns and also the types of Turing patterns generated by the network. The
paper [20] has experimentally studied the role of initial conditions in relation to
a type of bifurcations, named grazing bifurcations, for a fixed set of parameters,
in a memristor circuit with a nonsmooth memristor characteristic. Numerical
studies in the (v, i)-domain on the influence of initial conditions on dynamics and
bifurcations have been carried out also in [21–23]. The articles [24, 25] developed
a method in the (v, i)-domain to study bifurcations, and in particular bifurcations
without parameters, in cases where the DAE description of memristor circuits is
available but the SE description is not. However, it is not analytically investigated
how initial conditions are related to such bifurcations. Stability properties of
attractors, local and global bifurcations, and the role of the initial conditions have
been extensively investigated in [26] as well. We refer the reader to [27, 28], and
references therein, for other contributions along this line.
Overall, the quoted papers aim at highlighting peculiar phenomena observable
in memristor circuits as the coexistence of a huge number of different attractors
for a fixed set of circuit parameters [10]. Moreover, bifurcations due to changing
initial conditions, for a fixed set of circuit parameters, are highlighted. One
fundamental shortcoming of these contributions is that such phenomena are
investigated basically by numerical or experimental means, while an analytic
explanation is lacking.
In this chapter we have shown, by discussing some significant applications,
that FCAM is effective to analytically explain these initial-condition related
phenomena and is especially well suited to rigorously prove the coexistence
of infinitely many different attractors and the presence of bifurcations without
parameters in memristor circuits. Such an extremely rich and complex dynamic
scenario corresponds to a case referred to in the literature as extreme multistability. The reader is referred to [10] for other classes of nonlinear dynamical systems
in physics and engineering featuring extreme multistability.