Figure 1:
Schematic representation of the memristor composed of undoped (TiO) and doped oxygen vacancies (TiO) with platinum electrodes (Pt).
Figure 2:
Temporal evolution of (a) , (b) , (c) , and (d) . We consider the parameters according to Table 1. We observe periodic oscillations.
Figure 3:
as a function of for (a) nm, (b) nm, (c) nm, and (d) nm. Depending on , it is possible to observe the existence of a pinched hysteresis effect, as shown in the panels (b) and (c).
Figure 4:
Schematic diagram for the three circuit elements in series [12[12] B. Muthuswamy and L.O. Chua, Int. J. Bifurcation Chaos 20, 1567 (2010).]. The circuit is composed of a linear passive inductor (), a linear passive capacitor (), and a nonlinear active memristor ().
Figure 5:
Phase portraits of versus for , , , (a) , and (b) . The panels (a) and (b) show periodic and chaotic attractors, respectively.
Figure 6:
(a) Bifurcation diagram and (b) Lyapunov exponents as a function of the control parameter for , , and . The blue () and green () dashed lines correspond to the values considered in Figs. (a) and (b), respectively.
Figure 7:
Schematic diagram for the Chua circuit with a non ideal voltage controlled memristor [15[15] K. Rajagopal, S. Kacar, Z. Wei, P. Duraisamy, T. Kifle and A. Karthikeyan, Int. J. Electron. and Commun 107, 183 (2019).]. The memristor replaces the Chua diode.
Figure 8:
Phase portraits of versus showing the coexistence of attractors for , , , , , (a) , and (b) . The panel (a) displays three periodic attractors, while the panel (b) exhibits a periodic (in red) and a chaotic attractor (in green).
Figure 9:
Bifurcation diagram of the local maximum of the dynamical variable as a function of the control parameter for , , , , and . The blue and black lines mark the values used in the phase portraits of Figs.(a) and(b), respectively.
Figure 10:
Basins of attraction with for attractors depicted in Fig.(a) () and Fig.(b) (). The white, blue, and red regions correspond to subsets of the initial conditions of different attractors with period one, while the green region corresponds to a chaotic attractor.
Figure 11:
Schematic diagram for the Colpitts circuit where BJT is a bipolar junction transistor [36[36] G.M. Maggio, O. De Feo and M.P. Kennedy, IEEE Trans. Circuits Syst. I 46, 1118 (1999).]. The circuit has a BJT as the gain element and a resonant network composed of a pair of capacitors ( and ) and an inductor ().
Figure 12:
versus for , , , and . The panels show the coexistence of many attractors, where we consider as initial conditions (a) () (point attractor in black star) and () (period 1 limit cycle), (b) () (period 2 limit cycle), (c) () (period 3 limit cycle), (d) () (asymmetric chaotic double-scroll attractor), (e) () (symmetric chaotic double-scroll attractor), and (f) () (unbounded orbit).