Slow–Fast Dynamics in a Chaotic System with Strongly Asymmetric Memristive Element

We investigate the effect of a memristive element on the dynamics of a chaotic system. For this purpose, the chaotic Chua’s oscillator is extended by a memory element in the form of a double-barrier memristive device. The device consists of [Formula: see text]/Al2O3/Al/Nb layers and exhibits strong analog-type resistive changes depending on the history of the charge flow. In the obtained system we observe strong changes in the dynamics of chaotic oscillations. The otherwise fluctuating amplitudes of Chua’s system are disrupted by transient silent states. Numerical simulations and analysis of the extended model reveal that the underlying dynamics possesses slow–fast properties due to different timescales between the memory element and the base system. Furthermore, the stabilizing and destabilizing dynamic bifurcations are identified that are traversed by the system during its chaotic behavior.


Introduction
The field of nonlinear dynamics, chaos, and complexity has attracted increasing interest from the point of fundamental science and engineering during the last decades [Schuster & Just, 2005;Strogatz, 2015]. Classical and well-explored nonlinear phenomena form a fundamental scientific repertoire to shed more light on novel interdisciplinary research areas such as complex network systems. To name but a few, this includes spatiotemporal pattern formation in chemical reactions, pulse coupled oscillators, chaotic weather formation or timedelay systems [Epstein & Showalter, 1996;Pikovsky et al., 2003;Yanchuk & Giacomelli, 2017]. Currently, time-varying networks, neuroscience, and social dynamics are areas of intense research efforts in nonlinear science [Sporns, 2011;Osipov et al., 2007;Skarda & Freeman, 1987;Buzsaki, 2006]. Furthermore, nonlinear systems with experimentally observable chaotic signatures have received much attention, as they are widely distributed over many different fields including optical, mechanical and chemical systems [Epstein & Showalter, 1996;Gilet & Bush, 2009;Ievlev et al., 2014;Brzeski et al., 2015;Buzsaki, 2006]. In this context, electronic systems are of particular interest. Rather simple analog circuits allow the study and control of chaos and nonlinear dynamical phenomena. The fast and easy access to system parameters in experiments through the variation of passive elements of the circuit, i.e. resistances, inductances and capacitances, is an effective way to tune the circuit dynamics and to observe the results in real time. The first chaotic circuit was realized by Leon Chua in the 1980s, consisting of three energy storing elements and a nonlinear electronic device. The circuit exhibits a classical period-doubling route to chaos as well as a chaotic double-scroll attractor [Matsumoto, 1984;Chua et al., 1993].
In this paper, we present a realization of Chua's circuit comprising a semiconductor device which shows memrisitve properties [Vaidyanathan & Volos, 2017]. Memristive devices are currently investigated from the perspective of nonvolatile memories and promising devices to mimic basal synaptic mechanisms in neuromorphic circuits [Ielmini & Waser, 2016;Tetzlaff, 2014;Kozma et al., 2012;Adamatzky & Chua, 2014]. In general, a memristive device connects the current I and voltage V nonlinearly. The resistance of such a system depends on a mechanism relating the voltage to a change of an internal state z: In its simplest form such a device consists of a metal-insulator-metal capacitor-like structure.
Here, an applied voltage can lead to the movement of ions within the insulator, resulting in a change of the resistance [Strukov et al., 2008]. Thus, the history of the applied voltage is connected to the current state of the device. As a result, the currentvoltage characteristics or I-V curve of a memristive device exhibits a hysteresis loop. For more details about memristive devices and the underlying physical and chemical mechanisms, see [Ielmini & Waser, 2016].
Recently, a variety of interesting dynamics evoked by the implementation of a memristive device into autonomous oscillators have been found and proposed. The observed dynamics range from the addition of multiple scrolls into a chaotic system [Wang et al., 2017], multiple wings in a 4D system [Zhou et al., 2016;Zhou et al., 2018b] to various attractors in a memrisitve augmented Twin-T Oscillator [Zhou et al., 2018a]. Furthermore, also techniques for the analyses of such systems have been developed further. These include the Flux-Charge Analysis technique which can have advantages regarding the description of the system and its numerical solution [Corinto & Forti, 2017]. However, due to limited availability of robust memristive devices, nearly all approaches have in common that they rely on idealistic theoretical memristive models as well as sophisticated circuits mimicking their behavior. Nonetheless, nanoscale memristive devices are promising candidates to improve energy efficiency and increase the integration density of electronic systems. Therefore, in this article we investigate experimentally and theoretically the effects of a more realistic memristive device with asymmetric characteristic curve. Starting from experimental observations (Sec. 2) we propose a detailed mathematical model (Sec. 3) and analyze its dynamics (Sec. 4). In particular, we show how the slow-fast chaotic oscillations emerge.

Experiment
Figure 1(a) shows the setup considered in this work. The chaotic circuit as proposed by Leon Chua is extended with a memristive device in parallel to Chua's diode. Therefore, the solid state device superimposes the necessary nonlinearity to drive the chaotic circuit. The current flow through the device discharges the capacitor resulting in an additional negative feedback on it. As the state of the double-barrier memristive device (DBMD) depends strongly on the history of the applied voltages across it, the strength of the additional discharge of the capacitor varies chaotically with time. In Fig. 1 memristive device is shown. The internal structure of the device consists of a Au/NbO x /Al 2 O 3 /Al/Nb layer sequence [Hansen et al., 2015;Dirkmann et al., 2016]. The pinched hysteresis, which is characteristic for memristors, is well visible. This means, that the application of a positive voltage changes the state of the device. As a result, the measurable current flow through the device changes depending on the history of the applied voltage. This transition from a high-resistance state (HRS) to a lowresistance state (LRS) becomes visible by applying a positive voltage to the Au electrode in respect to the Nb electrode. The device remains in the LRS after the polarity of the applied voltage switches, but changes its resistance again in an analog fashion to a HRS, if the voltage is beyond a threshold voltage [Hansen et al., 2015]. We emphasize that this kind of device is filament-free, i.e. an interfacebased switching is responsible for the pinched hysteresis observed in the I-V curve. Furthermore, the switching is not binary but continuous. Since the change in the state of the memristive element in use is sensitive to the duration of the applied voltage, the intrinsic frequency of Chua's oscillator needs to be adapted. Therefore, the inductance and capacities need to be fairly big to decrease the frequency into the single Hz range. A common strategy in building a huge inductance experimentally is to use a gyrator. In Fig. 1(c) the resulting experimental setup is depicted (for details on the parameters of the devices, see Appendix A). Besides the inductance, Chua's diode is built with active elements as well. Two negative impedance converters built in parallel show the characteristic curve of Chua's diode. The diode is necessary for the autonomous oscillations of the chaotic oscillator. In Fig. 1(d) a photograph of the experimental setup is shown. The small chip with double-barrier memristive devices is located in the top right of the printed circuit board. The experimental results from the circuit shown in Fig. 1(c) are depicted in Fig. 2. In (a) a long time series measurement of the augmented system is shown. Besides the chaotic oscillations of Chua's oscillator, the system shows transient episodes of local dampening. These episodes occur irregularly during the time evolution of the system. In Fig. 2(b) an enlargement around one of these episodes is shown. The transient dampening is highlighted in red. In the (v 1 , v 2 ) projection of the phase space, the transition to an additional attractor is well visible. The reduction in the oscillation amplitude refers to the transition from the original chaotic attractor to the part of the attractor introduced by the DBMD. In the following, we will present a model to describe and analyze the observed dynamics in the augmented system.

Model
Since the memristive device is implemented experimentally in parallel to Chua's diode, the voltage drop across the device is equal to the state variable v 1 of the original chaotic oscillator. Therefore, the current flow through the DBMD discharges the capacitor and functions as a negative feedback to the first state-variable v 1 . The modified equations of Chua's system augmented by the memory element are as follows: with Chua's diode modeled as a piecewise-linear function: The DBMD consists of a multilayer structure of different materials [see Fig. 1(b)], which can be interpreted in terms of an equivalent circuit as depicted in Fig. 3. The metal semiconductor transition is modeled as a Schottky contact followed by a tunnel barrier. Thus, the equations describing the equivalent circuit are as follows: where v e and v t are internal voltage variables as shown in Fig. 3. The memory component of the DBMD is represented through the state variable z, which refers to the average ion-position inside the active layer, that is the NbO x solid-state electrolyte. During the switching oxygen vacancies move and consequently decrease and increase the potential on the interface at the Schottky contact and the tunnel barrier, respectively [Solan et al., 2017]. The voltage over the Schottky contact is v s = v 1 − v e − v t and leads to the total current i s (v s , z) through the device as: with ϕ s (z) as the state-dependent normalized Schottky-barrier height, n(z) as an ideality factor and α f as a fitting parameter for the Schottkyeffect denoted by the normalized Schottky-barrier thickness α s . The amplitude of the current I s scales the total current depending on the temperature and device area, respectively [Solan et al., 2017].
The closed system (2)-(5) describes the dynamics of the complete circuit augmented with a DBMD (for additional information about the model see Appendix B). As one can observe from Fig. 4, the results of the numerical simulation of the closed system are in agreement with the measurement data (compare Fig. 2). The purely chaotic dynamics are interrupted by relatively long time intervals of almost constant voltages and currents. After a period nearly without any oscillations, an onset of the local oscillations follows. Again, the local oscillations are amplified until the trajectory switches to the opposite side of the characteristic double-scroll attractor. The strong diode-like characteristic of the DBMD diminishes the influence of itself on the other side of the chaotic double-scroll attractor. This results in the asymmetric change of the system behavior. The numerical solution demonstrates clearly the role of the internal state variable of the DBMD.
In Fig. 5(a) the time series for one typical transient silent state (TSS) is shown in the (v 1 , v 2 , z) projection of the phase space. The decrease of the internal state z can be observed over time, which leads to a higher conductance of the device and a damping of the oscillations. It is followed by a steep T. Birkoben et al. increase when the resuming chaotic oscillations lead to a negative voltage over the memristive device. The derivative of z is depicted in Fig. 5.

Slow-Fast Motion
In the following, we show that the chaotic dynamics with TSS events can be understood as a slow-fast motion with the slowest timescale governed by the memory z. More specifically, the variable z can be considered as a control parameter for the remaining faster variables, i.e.ż = 0 [Dumortier et al., 1996;Krupa et al., 1997;Desroches et al., 2012;Kuehn, 2015;Jardon-Kojakhmetov & Kuehn, 2019]. Figure 6(a) shows a bifurcation diagram for v 1 in dependence on z as a control parameter. For the experimentally chosen value R = 1.6 MΩ, the fast dynamics possess a stable stationary state (v e (z), v t (z), v 1 (z), v 2 = 0, i 3 (z)) for all values of z in the interval 0.71 < z < 0.86, shown by the solid blue line in Fig. 6(a) stable critical manifold [Kuehn, 2015], and the trajectories of the full system are attracted to this manifold on the fast timescale. Being close to the manifold, the dynamics are then governed in Fig. 6(a)] is called the by the single scalar equation (4c) for the z variable with all fast variables being confined to the manifold. With v 1 > 0 Eq. (4c) reduces to: as v r (v 1 > 0, v s , z) = 0 and ϕ a (v 1 > 0, z) = ϕ a1 + z(ϕ a0 − ϕ a1 ). From the differential equation (6) it can be seen that z decreases on the slow manifold as positive values for v 1 lead to a negative sign of the derivative [cf. Fig. 6(b)]. This corresponds to motion along the manifold to the left. These multiscale arguments explain parts of the dynamics observed in the full system and are shown in Fig. 6(a). Once the trajectory is attracted to the slow manifold [point A in Fig. 6(a)], the TSS episode starts and the memory variable z decreases slowly until it reaches the Hopf point HP 2 at z = 0.71. Beyond this point, the slow manifold becomes unstable and the trajectory exhibits amplified oscillations, while z still decreases further. When the oscillations become sufficiently large, the orbit leaves the neighborhood of the slow manifold and the TSS episode ends. In fact, we observe here a classical delayed stability loss, when the orbits remain for certain time close to the unstable part of the slow manifold [Dumortier et al., 1996;Krupa et al., 1997;Desroches et al., 2012;Kuehn, 2015;Jardon-Kojakhmetov & Kuehn, 2019].
After the TSS has terminated, the voltages v 1 , v e , and v t decrease and the variable z accelerates and approaches the fast timescale [cf. Fig. 5(b)] for the gradientż along the trajectory. In this way, a slow setting to a LRS and a faster reset to a HRS drives the slow-fast dynamics behind the TSS.
Figure 6(c) shows the positions of the Hopf points on the slow manifold. These determine the length of the TSS phases, depending on R. See for instance, the intersection of the dashed line at R = 1.6 MΩ with the stable area marked in blue. The stable branch can be seen to expand with increasing R, until the stability becomes independent of z as HP 1 and HP 2 collide. For decreasing R, the stable branch shrinks and vanishes with HP 2 and HP 3 .

Conclusion
In summary, we have described the influence of a new solid state memory device on a model system exhibiting chaotic dynamics to study the influence of memory on chaos. Interestingly, the intrinsic memory of the DBMD has a stabilizing and ordering effect on the otherwise purely chaotic motions of the system. The experimental observation as well as the subsequent theoretical treatment reveal the underlying dynamics. We have observed that the memory acts on the slowest timescale, and thus, it can be considered as an intrinsic slowly-changing bifurcation parameter. In particular, when the fast chaotic motion of the system is interrupted by a TSS, the dynamics can be considered as a steady state, which is adiabatically changing with the slow variation of the memory. Then, the chaotic oscillations are dampened until the stability changes. Such a change occurs when the memory reaches a threshold, and a Hopf bifurcation of the fast system leads to amplification of chaotic oscillations. As the memristive device in use has two timescales for the set and reset of the resistive states, this damping and successive amplification is driven by these slow-fast dynamics. Although this study is restricted to an electronic system, the observed behavior and general dynamical changes might be found in a broader range of systems. The occurrence of transiently stable and ordered behavior in otherwise highly nonlinear or chaotic trajectories might be related to dynamic bifurcations as an intrinsic memory element of these systems changes its state. The sequence of these transient states is primarily driven by the underlying chaotic motions but the duration and recovery depend strongly on the characteristic timescales of this memory element. Speaking in more general terms it is driven by the time needed to adapt to a new input. The considered model is a practically important example of a system with slow-fast dynamics, where such theoretical multiscale methods as singular perturbation theory or averaging [Dumortier et al., 1996;Krupa et al., 1997;Desroches et al., 2012;Kuehn, 2015;Jardon-Kojakhmetov & Kuehn, 2019] can be further applied for more detailed analysis. decrease the occurring currents, the resistance R is increased. The huge inductance is implemented experimentally through a gyrator. The recordings of the state variables are done using a PicoScope 3000 Series. Furthermore, to decrease the influence of the measurement on the circuit, unity gain followers are built with JFET general purpose operational amplifiers (TL074 and TL071). The simulations are carried out using Python 3.6 running the scipy library.

Appendix B Additional Information about the Model
The following set of equations describes the ion motions inside the DBMD as well as the increase and decrease of the interfacial energy barrier heights: ϕ s (z) = ϕ s0 + z(ϕ 1 − ϕ 0 ), (B.1d) n(z) = n 0 + z(n 1 − n 0 ), (B.1e) The used values for the simulation are as follows: