Bifurcation without parameters in a chaotic system with a memristive element
Tom Birkoben, Moritz Drangmeister, Finn Zahari, Serhiy Yanchuk, Philipp Hövel, Hermann Kohlstedt
BBifurcation without parameters in a chaotic system with a memristive element
Tom Birkoben, ∗ Moritz Drangmeister, † Finn Zahari, ‡ Serhiy Yanchuk, § Philipp H¨ovel,
4, 2, ¶ and Hermann Kohlstedt ∗∗ Nanoelektronik, Technische Fakult¨at, Christian-Albrechts-Universit¨at zu Kiel, Kaisertraße 2, 24143 Kiel, Germany Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany Institut f¨ur Mathematik, Technische Universit¨at Berlin,Straße des 17. Juni 136, 10623 Berlin, Germany School of Mathematical Sciences, University College Cork, Western Road, Cork T12 XF64, Ireland (Dated: June 7, 2019)We investigate the effect of memory on a chaotic system experimentally and theoretically. Forthis purpose, we use Chua’s oscillator as an electrical model system showing chaotic dynamicsextended by a memory element in form of a double-barrier memristive device. The device consistsof Au/NbO x /Al O /Al/Nb layers and exhibits strong analog-type resistive changes depending onthe history of the charge flow. In the extended system strong changes in the dynamics of chaoticoscillations are observable. The otherwise fluctuating amplitudes of the Chua system are disruptedby transient silent states. After developing a model for Chua’s oscillator with a memristive device,the numerical treatment reveals the underling dynamics as driven by the slow-fast dynamics of thememory element. Furthermore, the stabilizing and destabilizing dynamic bifurcations are identifiedthat are passed by the system during its chaotic behavior. The field of nonlinear dynamics, chaos, and complexityhas attracted increasing interest from the point of funda-mental science and engineering during the last decades[1, 2]. Classical and well-explored nonlinear phenomenaform a fundamental scientific repertoire to shed morelight on novel interdisciplinary research areas such ascomplex network systems. To name but a few, this in-cludes spatio-temporal pattern formation in chemical re-actions, pulse coupled oscillators, chaotic weather forma-tion or time-delay systems [3–5].Currently, time-varying networks, neuroscience, and so-cial dynamics are areas of intense research efforts innonlinear science [6–9]. Furthermore, nonlinear systemswith experimentally observable chaotic signatures havereceived much attention, as they are widely distributedover many different fields including optical, mechanicaland chemical systems [3, 9–12]. In this context, electronicsystems are of particular interest. Rather simple analogcircuits allow the study and control of chaos and nonlin-ear dynamical phenomena. The fast and easy access tosystem parameters in experiments through the variationof passive elements of the circuit, i.e., resistances, induc-tances and capacitances, is an effective way to tune thecircuit dynamics and to observe the results in real time.The first chaotic circuit was realized by Leon Chua in the1980s, consisting of three energy storing elements and anonlinear electronic device. The circuit exhibits a clas-sical period-doubling route to chaos as well as a chaoticdouble-scroll attractor [13, 14].In this letter, we present a novel realization of Chua’scircuit comprising a memristive device, i.e., a storageelement [15]. Memristive devices are currently investi-gated from the perspective of non-volatile memories andpromising devices to mimic basal synaptic mechanismsin neuromorphic circuits [16–19].In general a memristive device connects the current I and voltage V nonlinearly.The resistance of such a system depends on a mechanismrelating the voltage to a change of an internal state z : I = G ( z ) · V, (1a)˙ z = f ( z, V ) . (1b)In its simplest form such a device consists of a metal-insulator-metal capacitor-like structure. Here, an appliedvoltage can lead to the movement of ions within the insu-lator, resulting in a change of the resistance [20]. Thus,the history of the applied voltage is connected to the cur-rent state of the device. As a result, the current-voltagecharacteristics or I-V curve of a memristive device ex-hibits a hysteresis loop. For more details about mem-ristive devices and the underlying physical and chemicalmechanisms see Ref. [16].Figure 1(a) shows the circuit layout considered in thisletter. The chaotic circuit as proposed by Leon Chua isextended with a memristive device in parallel to Chua’sdiode. Therefore, the solid state device superimposes thenecessary nonlinearity to drive the chaotic circuit. As thestate of the double-barrier memristive device (DBMD)depends strongly on the history of the applied voltagesacross it, the strength of the additional discharge of thecapacitor varies chaotically with time. The used de-vice consists of a Au/NbO x /Al O /Al/Nb layer sequence[22, 23]. We emphasize that this kind of device is fila-ment free, i.e., an interface-based switching is responsi-ble for the pinched hysteresis observed in the I-V curve.Furthermore, the switching is not binary but continuousas depicted in Fig. 1 (b). The transition from a high-resistance state (HRS) to a low-resistance state (LRS)is well observable for applying a positive voltage to theAu electrode in respect to the Nb electrode. The deviceremains in the LRS after the polarity of the applied volt- a r X i v : . [ n li n . C D ] J un (a) i v D B M D C hu a ' s D i od e Au Al O Al Nb (b)(c) VI V (V) | I | ( A ) V ( V ) t (s) ve vt Schottky contact tunnel barrier is Ce Ct
Re Rt(vt, z)
NbO x R v L i s FIG. 1. (a) Experimental setup of Chua’s circuit compris-ing a double-barrier memristive device (DBMD). The deviceis integrated in parallel to Chua’s diode. The current flowthrough the device discharges the capacitor resulting in anadditional negative feedback on it. (b) A typical
I-V curveof a DBMD. It consists of Au/NbO x /Al O /Al/Nb thin lay-ers. As a triangular voltage (inset) is applied to the device,the current increases. This changes the internal state of thedevice leading to a transition from a high-resistance state toa low-resistance state. Applying a negative voltage resets thelow-resistance state again to a high-resistance state. We liketo emphasize that this transition is not binary but continu-ous. (c) Equivalent circuit of the DBMD. It can be subdividedinto a Schottky contact and a tunnel barrier. The resistancechanges as the applied voltage modulates the effective barrierheights of the tunnel barrier and the Schottky contact, re-spectively. For further details on the derivation of the modelsee Ref. [21]. age switches, but changes its resistance again in an analogfashion to a HRS after a threshold voltage was exceeded[22]. Considering the internal structure of a DBMD thefunctional films can be modelled as an equivalent circuitas shown in Fig. 1 (c). The metal semiconductor transi-tion is modeled as a Schottky contact followed by a tunnelbarrier. From this model the following set of differentialequations, which describe the state of the DBMD, can be obtained: [21]˙ v e = 1 C e (cid:18) i s ( v s , z ) − v e R e ( z ) (cid:19) , (2a)˙ v t = 1 C t ( i s ( v s , z ) − i t ( v t , z )) , (2b)˙ z = − ˆ Zω ( z ) e ϕ a ( v ,z ) sinh (cid:18) v r ( v , v s , z ) + v e − V c V e (cid:19) . (2c)The memory component of the DBMD is representedthrough the state variable z , which refers to the aver-age ion-position inside the active layer, that is the NbO x solid-state electrolyte. During the switching oxygen-vacancies move and consequently decrease and increasethe potential on the interface at the Schottky contact andthe tunnel barrier, respectively [21]. The voltage over theSchottky contact is v s = v − v e − v t and leads to the totalcurrent i s ( v s , z ) through the device as: i s ( v s , z ) = I s exp − ϕ s ( z ) − α f (cid:115) | v s | − v s α s V ϑ × (cid:20) exp (cid:18) v s n ( z ) V ϑ (cid:19) − (cid:21) . (3)with ϕ s ( z ) as the state-dependent normalized Schottky-barrier height, n ( z ) as an ideality factor and α f as afitting parameter for the Schottky-effect denoted by thenormalized Schottky-barrier thickness α s . The ampli-tude of the current I s scales the total current dependingon the temperature and device area, respectively [21].Since the memristive device is implemented experimen-tally in parallel to Chua’s diode, the voltage drop acrossthe device is equal to the state variable v of the originalchaotic oscillator. Therefore, the current flow throughthe DBMD discharges the capacitor and functions as anegative feedback to the first state-variable v . The mod-ified equations of the system augmented by the memoryelement are as follows:˙ v = 1 C (cid:18) v − v R − f ( v ) − i s ( v s , z ) (cid:19) , (4a)˙ v = 1 C (cid:18) v − v R − i (cid:19) , (4b)˙ i = − L v , (4c)with Chua’s diode modelled as a piece-wise linear func-tion: f ( v ) = m v + m − m | v + B p | − | v − B p | ) . (5)The closed system (2a)-(4c) describes the dynamics ofthe complete circuit augmented with a DBMD (addi-tional information in [24]).As one can observe from Fig. 2, the additional memory FIG. 2. (a) Snapshot of the experimental time series measurement of Chua’s circuit comprising a DBMD. The top three tracesshow the time evolution of the fundamental state variables of the system. The chaotic oscillations (grey) are disrupted byminute long dampening of the intrinsic oscillations (red). The trajectory of the system will leave this transient silent state andreturn to the well known chaotic oscillations. The lower diagram shows the ( v , v ) phase space. (b) Snapshot of the numericalresults of the extented chaotic system comprising a DBMD. Damping and concurrent excitation of the Chua variables v , v and i can be observed as well and are in good agreement to the experimental results (additional information in [24]). element has a significant influence on the system dy-namics. In the original system without the memristivedevice, the local oscillations are amplified until thetrajectory switches to the opposite side of the charac-teristic double-scroll attractor. With the introduction ofthe memristive device the purely chaotic dynamics areinterrupted by transient silent states (TSS). These statesare characterized by the damping and relatively longtime-intervals of almost constant voltages and currents.After a period nearly without any oscillations, an onsetof the local oscillations follows. The strong diode-likecharacteristic of the DBMD diminishes the influenceof the it on the other side of the chaotic double-scrollattractor. This results in the clearly asymmetric changeof the system behavior.The numerical solution shows clearly the role of theinternal state variable of the DBMD. In Fig. 3 the timeseries for one typical TSS is shown in the ( v , v , z )phase space. The decrease of the internal state z can beobserved over time, which leads to a higher conductance FIG. 3. Trajectory of the chaotic system, exhibiting a tran-sient silent state by lowering the resistance and the successivesetback to a high-resistance state. The evolution starts at thegreen diamond and stops at the blue one. The decrease of z is related to the onset of the dumping of the oscillations. of the device and a damping of the oscillations. Itis followed by a steep increase when the resumingchaotic oscillations lead to a negative voltage over thememristive device. FIG. 4. (a) Bifurcation diagram for v over control param-eter z with stationary solutions in blue, periodic solutionsin orange and chaotic solutions in purple. Continuous linesdepict stable solutions, dashed lines unstable ones. For peri-odic and chaotic attractors, the minimum and maximum of v are shown. Superimposed to the bifurcation diagram is atransient silent state with a green diamond and blue diamondcorresponding to the starting and stopping point, respectively.The trajectory follows the arrows. (b) Position of the Hopfbifurcation points (HP) parametrized by R if z is varied asa control parameter. The blue shaded area shows where astable branch exists in the solution. During the experiment R is fixed at 1.6 MΩ (dashed, horizontal line). In the following we show that the chaotic dynamics withTSS events can be understood as a slow-fast motionwith the slowest timescale governed by the memory z .More specifically, the variable z can be considered as acontrol parameter for the remaining faster variables, i.e.,˙ z = 0 [25]. For the experimentally chosen value R = 1 . v e ( z ) , v t ( z ) , v ( z ) , v = 0 , i ( z )) for all values of z in theinterval 0 . < z < .
86, see solid blue line in Fig. 4(a).At the boundaries, z = 0 .
71 and z = 0 .
86, the branch becomes unstable in subcritical Hopf bifurcations. Thisset (solid blue line in Fig. 4(a)) is called the stable slowmanifold [25], and the trajectories of the full system areattracted to this manifold on the fast timescale. Beingclose to the manifold, the dynamics are then governedby the single scalar equation (2c) for the z variable withall fast variables being confined to the manifold. Fromthe differential equation (2c) of the internal state z , itcan be seen that z decreases on the slow manifold aspositive values for v , v e , and v t lead to a negative signof the derivative. This corresponds to motion along themanifold to the left. The above multi-scale argumentsexplain parts of the dynamics observed in the full systemand are shown in Fig. 4(a). Once the trajectory isattracted to the slow manifold (point A in Fig. 4(a)), theTSS episode starts and the memory variable z decreasesslowly until it reaches the Hopf point, HP , at z = 0 . z still decreases further. When the oscillations becomesufficiently large, the orbit leaves the neighborhood ofthe slow manifold and the TSS episode ends. Afterthe TSS has terminated, the voltages v , v e , and v t decrease and the variable z accelerates and approachesthe fast timescale, see Fig. 5 for the gradient ˙ z alongthe trajectory. In this way, a slow setting to a LRS anda faster reset to a HRS drives the slow fast dynamicsbehind the TSS. Figure 4(b) shows the changing size ofthe stable part of the slow manifold, this is the lengthsof the TSSs phases, depending on R . In particular, thestable branch can be seen to expand with increasing R ,until the stability becomes independent of z when HP and HP meet. For decreasing R , the stable branchshrinks and vanishes with HP and HP . FIG. 5. The gradient of z as a function of the state variable v along the trajectory. The red trace corresponds to the slowdecrease of z during the transient silent state event. The greypart refers to the setback. The green and blue diamond marksmark the start and the endpoint, respectively. In summary we have described the influence of a newsolid state memory device on a model system exhibitingchaotic dynamics to study the influence of memoryon chaos. Interestingly, the intrinsic memory of theDBMD has a stabilizing and ordering effect on theotherwise purely chaotic motions of the system. Theexperimental observation as well as the subsequenttheoretical treatment reveal the underlying dynamics.We have observed that the memory acts on the slowesttimescale, and thus, it can be considered as an intrinsicslowly-changing bifurcation parameter. In particular,when the fast chaotic motion of the system is interruptedby a TSS, the dynamics can be considered as a steadystate, which is adiabatically changing with the slowvariation of the memory. Then, the chaotic oscillationsare dumped until the stability changes. Such a changeoccurs when the memory reaches a threshold, and aHopf bifurcation of the fast system leads to amplificationof chaotic oscillations.As the memristive device in use has two time scalesfor the set and setback of the resistive states, thisdumping and successive amplification is driven by theseslow-fast dynamics. Although this study is restrictedto an electronic system, the observed behavior andgeneral dynamical changes might be found in a broaderrange of systems. The occurrence of transiently stableand ordered behavior in otherwise highly nonlinearor chaotic trajectories might be related to constantdynamic bifurcations as an intrinsic memory element ofthese systems changes its state. The sequence of thesetransient states is primary driven by the underlyingchaotic motions but the duration of and recovery fromeach depends strongly on the characteristic timescales ofthis memory element. Speaking in more general terms itis driven by the time needed to adapt to a new input.The authors acknowledge financial support byDeutsche Forschungsgemeinschaft (DFG, German Re-search Foundation), projects RU2093 (TB, FZ, HK),411803875 (SY) and Collaborative Research Center 910(MD, PH). ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected][1] H. G. Schuster and W. Just, Deterministic chaos: anintroduction , 4th ed. (Wiley-VCH, Weinheim, 2005).[2] S. H. Strogatz,
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