Route to logical strange nonchaotic attractors with single periodic force and noise
aa r X i v : . [ n li n . AO ] S e p Route to logical strange nonchaotic attractors with single periodic forceand noise
M. Sathish Aravindh,
1, 2, a) A. Venkatesan, b) and M. Lakshmanan c) PG & Research Department of Physics, Nehru Memorial College (Autonomous), Affiliated to Bharathidasan University,Puthanampatti, Tiruchirappalli - 621 007, India. Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli - 620 024,India.
Strange nonchaotic attractors (SNAs) have been identified and studied in the literature exclusively in quasiperiodicallydriven nonlinear dynamical systems. It is an interesting question to ask whether they can be identified with other typesof forcings as well, which still remains as an open problem. Here, we show that robust SNAs can be created by a smallamount of noise in periodically driven nonlinear dynamical systems by a single force. The robustness of these attractorsis tested by perturbing the system with logical signals leading to the emulation of different logical elements in the SNAregions.
The question whether strange nonchaotic attractors(SNAs) can occur typically in nonlinear dynamical systemsother than the quasiperiodically forced ones still remainsas an open problem. In this paper, we show that SNAs canbe generated by a small amount of noise in a periodicallydriven Duffing oscillator with a single force. Robustnessof the resulting SNA phenomenon can be verified by per-turbing the system with logical signals. It is interesting tonote that robust SNAs with logical behavior (logical SNA)and without logical behavior (standard SNA) are observedwith different input streams. The logical behavior in theSNA regime is robust in the presence of experimental noisetoo. Thus the present study paves the way to constructalternative computing in reliable and reconfigurable com-puter architecture.
I. INTRODUCTION
Strange nonchaotic attractor(SNA) is an attractor which hasa complicated geometry but its maximal Lyapunov exponentis not positive and therefore it will not exhibit sensitive de-pendence on initial conditions. Such attractors were first ob-served by Grebogi , and since then the study of it has becomean active area of research in nonlinear dynamics. These at-tractors are realized in many theoretical models and experi-ments. Apart from observations of these attractors in typi-cal nonlinear systems such as logistic map, circle map, Duff-ing oscillator and van der Pol oscillator under quasiperi-odic forcing, these exotic attractors have also been experi-mentally observed in a quasiperiodically driven magnetoe-lastic ribbon system , in electronic circuits , in a plasmasystem , in an electrochemical cell and in a system near thetorus-doubling critical point . The evidence for these attrac-tors have been recently realized in the pulsation of stars like a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]
KIC 5520878 , in a nonsmooth dynamical system , a cel-lular neural network , and the quasiperiodically driven geo-physical Saltzman model .The study of SNA is of particular physical interest ina quantum particle in a spatially quasiperiodic potential .SNAs are important in biological systems too and forcommunication as well . Many of the previous studiesfocused on the routes and mechanisms by which an SNA isgenerated from a regular attractor or disappears as a chaoticattractor . Different routes to SNAs have been estab-lished in different nonlinear dynamical systems. In particular,routes like Heagy-Hammel route , fractilization of torus ,blowout bifurcation and intermittency and other routeshave been reported . SNAs are also characterized by var-ious tools including finite time Lyapunov exponents, phasesensitivity exponents, 0-1 test, recurrence plots, spectral dis-tribution and so on . Mathematically related issueshave also been addressed corresponding to the generation andproperties of SNA .Since many of the physically realized nonlinear dynami-cal systems do not fall under the category of quasiperiodicforcing, it is natural to question whether it is possible to real-ize SNAs in nonlinear systems with other types of forcings .It was shown in the literature that two asymmetrically cou-pled driven ring maps and a periodically driven oscillator withan inertial nonlinearity can produce SNAs via band mergingcrisis . Later, it was proved that these attractors are actuallychaotic . Aperiodic nonchaotic attractors have been createdin nonlinear systems using periodic forcing of high period .Many works have also focused on the generation of SNAs viastochastic forcing . It was suggested that a chaoticattractor can be converted into an SNA by adding suitablenoise . Later it was proved that the effect of noise smearsout the fine structure of the attractor and gives rise to negativeLyapunov exponents. Thus, the dynamics of this case is nei-ther strange nor nonchaotic . Wang et al. reported that robustSNAs can be induced by noise in autonomous discrete-mapsand in periodically driven continuous systems . They haveshown that in the periodic window, if the strength of noise sat-isfies the condition that D > D m , a critical value, the trajectoryof the system switches intermittently between the periodic at-tractor and the chaotic saddle . The Lyapunov exponent re-mains negative for noise amplitude D m < D < D ∗ m , where D ∗ m is the noise amplitude for which the maximal Lyapunovexponent is zero. In this range, the attractor of the system has astrange geometry but the maximal Lyapunov exponent is non-positive . They termed this attractor as a noise-inducedstrange nonchaotic attractor. Further, such SNAs have oc-curred in a small region of the parameter space.Thus, a very basic question arises : can it be shown thatSNAs exist in dynamical systems other than quasiperiodicallyforced ones and whether they are robust? Here robust SNA isreferred to the attractor to which an arbitrarily small changeof the system can not cause its destruction . Robustnessof SNA is essentially connected to its observation in exper-iments. Hunt and Ott have established analytically that in asimple mapping robust SNAs are generic under quasiperiodicforcing . They have shown that such robust strangenessof the attractors can be verified by calculating the informa-tion dimension (which has the value ‘1’) and capacity di-mension (which should be two for the mapping considered).Such an analysis is employed for a few more examples .Most of the studies of SNAs relied on computational verifica-tion. Even numerical approach has its own limitations. It waspointed out that the finite precision of calculation of computerleads to numerical errors .In the present paper, we address these issues by examin-ing the behavior of a periodically driven nonlinear oscillatorin the presence of noise. Specifically, we report the existenceof SNA in an optimal range of noise strength in a periodicallydriven double-well Duffing oscillator with a single force. Tovalidate the robustness of SNA, we perturb the system withlogical signals and examine whether this perturbation can al-ter the existence of SNAs in the system. We find that the SNAspersist in the noise induced periodically driven nonlinear sys-tem, even under perturbation.It is well known that noise is inevitable in many physi-cal situations and in fact it puts an upper limit on the per-formance of the system. In particular, the factor of noise isthe main concern as well as the limiting factor in the design-ing of digital integrated circuits and ultimately computer ar-chitecture. Although many nonlinear dynamical computingsystems have been proposed to make computing robust, reli-able and reconfigurable , the ambient noise and practicalnonidealities restrict one to emulate different logic elements.In this regard, recently the present authors have demonstrateda route to logical SNA in quasiperiodically driven nonlinearsystems . They showed that if the quasiperiodically drivenDuffing oscillator were perturbed by two logic signals, theoutput of the system reproduces logical behavior. They fur-ther demonstrated that by using such robust SNAs, one canemulate different logic gates in the presence of noise .In principle, it is possible to generate and maintainquasiperiodicity in a simple way but in practice it is difficult tocarry this out. For quasiperiodicity, forcing may be given withan irrational frequency or can be generated with two sourceswhose frequencies are incommensurate. Experimental uncer-tainties can usually lead to deviations in the precision of mea-suring rational or irrational numbers. Thus a question arisesnaturally: Can logical SNA arise in the absence of quasiperi- odic forcing? In other words whether logical SNAs can arisein situations where the underlying system has other forcingdependencies. In the present work, we identify a route to logi-cal SNA induced by noise in periodically driven Duffing oscil-lator with a single force. We perturb the system by using twosquare waves in the presence of noise and establish that theperturbed attractor persists with SNA properties. We furthershow that the output of the system reproduces logic elementscontrolled by noise. When we change the threshold or biasingof the system, the response of the system changes from onelogical operation to another one.The paper is organized as follows. We discuss the basicaspects of periodically driven Duffing oscillator in Sec.II. Wepresent noise induced SNAs in the above periodically drivensystem in Sec.III. We also describe the route to logical SNAand deduce the effect of two logical inputs. We further dis-cuss how to deduce the probability of getting logical behaviorand then show how to implement different logical behavior inSec.IV. Finally, we summarize our results in Sec.V. II. PERIODICALLY DRIVEN DOUBLE-WELL DUFFINGOSCILLATOR
In the present work, a periodically driven double-well Duff-ing oscillator by a single sinusoidal force of the followingform is considered:˙ x = y , ˙ y = − α ˙ x − β ( x − x ) + F sin θ + ε + √ D ξ ( t ) + I , ˙ θ = ω . (1)Eq.(1) is assumed as the equation of motion for a particle ofunit mass in the potential well V ( x ) = β (cid:0) − x + x (cid:1) . Thesimplest experimental realization of Eq.(1) is a magnetoelasticribbon .The quantities F and ω in (1) are the amplitude and fre-quency of the external forcing, respectively. ε and ξ ( t ) cor-respond to the asymmetric bias constant input and Gaussianwhite noise of intensity D , respectively. I is the amplitude ofthe input square wave signal. In our study we vary ‘ F ’, theamplitude of forcing parameter. The parameters are fixed forour numerical calculations as α = . , β = . , ω = . ε = . in the absence of noise ( D = . ) . III. NOISE-INDUCED SNAS IN PERIODICALLY DRIVENDUFFING OSCILLATOR
In the absence of noise ( D = ) , bias ε = . ( I = ) , we vary the forcing parameter ‘ F ’. It is observed thatthe system (1) exhibits typical period-doubling route to chaosas shown in Fig.1.Now, we consider the periodic window in Fig.1 in the range0 . < F < . F m and F ∗ m be the parame-ter values at the beginning and end of the periodic window FIG. 1. Bifurcation diagram of F vs x(t) in the presence of bias ε = . δ = . D = . -1-0.5 0 0.5 1 10000 40000 70000 100000(a) x ( t ) T -1-0.5 0 0.5 1 10000 40000 70000 100000(b) x ( t ) T -1-0.5 0 0.5 1 10000 40000 70000 100000(c) x ( t ) T FIG. 2. Panels (a)-(c) show the Poincarè surface of section in the timeseries plane with different values of noise strength with (a) D = .
0, (b) D = . D = . F = . ε = . where the maximal Lyapunov exponent is negative. It is wellknown that at the end of the window a chaotic saddle coex-ists with periodic orbits and this leads to transient chaos .Now we fix the system parameter value at F = . λ = − . D > D m = . D m = . < D < D ∗ m = . [see Figs.2(b) for D = . D > . | X ( Ω , N ) | ∼ N γ , where γ = . -8 -7 -6 -5 -4 Noise inducedSNA Λ log D FIG. 3. Largest Lyapunov exponent Λ against the noise amplitude‘D’. -12-8-4 0-0.06 -0.04 -0.02 0 0.02 0.04 0.06(a) P ( , λ ) λ l og | X ( Ω , N ) | log N -50-20 10 40 70 0 30 60 90 120 150 180(c) I m X ReX
FIG. 4. Panel (a) shows the distribution of the finite-time Lyapunovexponent in the presence of noise with D = . | X ( Ω , N ) | vs N γ on logarithmic scale with γ = . F = . ε = . δ = . [see Fig.4(b)]. Further, the trajectories in the complex planeof (ReX, ImX) exhibit fractal behavior as shown in Fig.4(c),for fixed parameters F = . D = . ε = .
1, and in the absence of logic input δ = . FIG. 5. Poincaré section of the phase space under white noise of am-plitude D = . ε = . It is well known that noise can smear out any strange geom-etry of the underlying attractor in random dynamical systems.Romeiras et al. have pointed out that the fractal structureof chaotic attractor can be resolved even under noise by an-alyzing the snapshot attractors constructed from out of a largenumber of trajectories . Wang et al. have explored thesesnapshot attractors for resolving the strange geometry of noiseinduced SNAs . Following their work, we also examinethe snapshot attractors for the present system formed by alarge number of trajectories. In particular we obtain the snap-shot attractors by using a grid of 100 ×
100 initial conditionsuniformly distributed in the region − . ≤ ( x , y ) ≤ . − . Figs.5(a)and 5(b) represent, respectively, a single trajectory and itsblow-up. Here, the points of the single trajectory and its blow-up part show that the points are randomly distributed. How-ever, the snapshot attractors formed by 10,000 trajectories asshown in fig. 5(c) and it blow-up part as shown in Fig.5(d)exhibit apparently a fractal structure. IV. ROUTE TO LOGICAL SNA WITH SINGLE PERIODICFORCE AND NOISEA. Effect of three level square wave in noise induced SNA
Past studies revealed that SNAs are typical in quasiperiodi-cally driven nonlinear dynamical systems, and it is importantto inquire whether the behavior of SNA in the present case isrobust. When we say that the SNA is robust it means that thebehavior persists under sufficiently small perturbations. It isan established fact that all robust behaviors are also typical,however not vice-versa . Motivated by the above facts, weinvestigate whether the SNA observed in noisy, periodicallydriven nonlinear system is typical and robust. For this pur-pose, we perturb the system with logic signals and analyzewhether SNA persists or not.In particular, we analyze the response of the system (1)under the effect of a logic input signal I . Specifically, thesystem (1) is driven with a low/moderate amplitude logic in-put signal I , where, I = I + I with two square waves ofstrengths I and I encoding two logic inputs. The inputs canbe either 1 or 0, giving rise to four distinct logic input sets ( I , I ) : ( , ) , ( , ) , ( , ) and ( , ) . For a logical ‘1’, we set I = I = + δ , while for a ‘0’, we set I = I = − δ . Here δ represents the strength of the input signal. We also note thatthe input sets (0,1) and (1,0) correspond to the same input sig-nal I . As a result, the four distinct input combinations ( I , I ) reduce to three distinct values of I , namely − δ , , + δ ,corresponding to the logic inputs ( , ) , ( , ) or ( , ) , ( , ) ,respectively.The output of the system is determined by the state x ( t ) ofsystem (1); for example, if δ = . − . . I & I , whereas I = I + I is a three-level square wave form − . ( , ) , 0 correspondingto the input sets ( , ) / ( , ) and + . ( , ) [see Fig.6(c)]. The output is determined by thedynamical variable x ( t ) of the system (1). Specifically, if for x ( t ) < x ∗ , where x ∗ is a threshold value of x ( t ) , the responseof the system is assumed to be the logical ‘0’ and if x ( t ) > x ∗ , -0.3 0 0.3 100000 110000 120000(a) I -0.3 0 0.3 100000 110000 120000(b) I -0.6 0 0.6 100000 110000 120000(c) I Time
FIG. 6. Panels: (a)-(b) show the two different logic inputs I and I ,respectively. (c) shows a combination of two input signals I + I .Input I = I = + . ′ ′ and I = I = − . ′ ′ . The ’3’ level square waves with -0.6correspond to the input set (0,0), 0 for (0,1)/(1,0) set and +0.6 for(1,1) input set. the output of the system is considered to be the logical ‘1’.The value of the threshold is to be selected appropriately. Inthe present case the threshold value is chosen as x ∗ =
0. Asa result. the output of the system is considered as logical ‘1’,if the variable x ( t ) of system resides entirely positive. On theother hand for logical ‘0’ it resides entirely in the negativeregion of the phase space.In the presence of the logic input we observe the route tological SNA in the noise assisted periodically driven double-well Duffing oscillator. Initially we add two square waves oflogic inputs in the periodically driven system and by includinga moderate noise we find that the system exhibits two kindsof strange nonchaotic attractor namely i) logical SNA (whichexhibits logical behavior) and ii) standard SNA (which doesnot exhibit logical behavior) in an optimal window regime aswe point out below. B. Route to logical SNA
Now we fix the parameters - the strength of input signals as δ = .
3, bias value ε = .
1, and the amplitude of the forcing as F = . -2-1 0 1 2 10000 40000 70000 100000(a) x ( t ) T -2-1 0 1 2 10000 40000 70000 100000(b) x ( t ) T -2-1 0 1 2 10000 40000 70000 100000(c) x ( t ) T -2-1 0 1 2 10000 40000 70000 100000(d) x ( t ) T FIG. 7. Panels (a)-(d) show the Poincarè surface of section in thetime series plane for different noise levels D = . D = . D = . D = . δ = . ε = . F = . -0.6 0 0.6 100000 110000 120000(a) I -1.001.0 100000 110000 120000(b) x ( t ) -1.001.0 100000 110000 120000(c) x ( t ) Time
FIG. 8. Panel (a) shows a combination of two input signals I + I .Input I = I = + . ′ ′ and I = I = − . ′ ′ . Panels (b) & (c) represent the correspond-ing dynamical response of the system x(t) under periodic forcingfor different noise levels D = . D = . ε = . chaotic windows, for a small noise value the periodic attractorand the chaotic saddle are not connected dynamically.In this case, the attractor remains a periodic one despite theinclusion of noise [see Fig.7(a)]. For D > D critical = . -9-6-3 0 -0.06 -0.04 -0.02 0 0.02 0.04a(i) P ( , λ ) λ -6-3 0 -0.04 -0.02 0 0.02 0.04a(ii) P ( , λ ) λ l og | X ( Ω , N ) | log N 2 3 4 5 3 3.5 4 4.5 5b(ii) l og | X ( Ω , N ) | log N-400-300-200-100 0 0 50 100 150 200c(i) I m X ReX 0 100 200 300-400 -300 -200 -100 0c(ii) I m X ReX
FIG. 9. Projection of the logical and standard SNA attractors ofEq.(1). Panels a(i) & a(ii) show the distribution of the finite-timeLyapunov exponent in the presence of different noise strengths, D = . D = . | X ( Ω , N ) | vs N γ on loga-rithmic scale for the SNAs with noise strengths D = . γ = . D = . γ = . b ( i ) & b ( ii ) for different noise strengths for fixed parameters F = . ε = . δ = . Fig.7(b)]. Since the chaotic saddle is part of the attractor, itis obvious that the attractor is strange and fractal. On fur-ther increase of D , it is observed that the intermittent visitsof the trajectory in the chaotic saddle region increases [seeFigs.7(c) & 7(d)]. As a result the maximal Lyapunov expo-nent keeps increasing from negative to positive value. The ex-ponent still remains negative as long as D < . D > . .Now, we analyze the above dynamics from the logical re-sponse point of view. When the input signal I = I + I iseither (1,1) or (0,1)/(1,0) states, it is found that the attractorresides in the x > x < x > x < D to D = . C. Characterization of noise induced logical SNAs andstandard SNAs
To characterize further that the attractor exhibits logical be-havior with SNA, we utilize the characterizations in terms offinite time Lyapunov exponents, Fourier power spectrum andfractal walks. The distributions of the FTLE with the forcingparameter F = . D = . D = . | X ( Ω , N ) | ∼ N γ , where γ = . γ = . (ReX, ImX) plane are demonstratedfor logical and standard SNAs [see Figs.9c(i) and 9c(ii)]. Allthe characterizations clearly indicate the noise-induced logi-cal and standard SNAs for the Duffing oscillator.Noise-induced SNAs and fractal snapshot attractors canalso occur due to the effect of logic signals in the periodicallydriven Duffing oscillator system. We have again constructedthe snapshot attractor as discussed in Sec.III for this case too.It is demonstrated clearly in Figs.10(a) and (b) of a single tra-jectory of the logic SNA and standard SNA respectively, donot reveal the fractal structure while the snapshot attractors[see Figs.10(c) and (d)] are apparently fractal. D. Probability of obtaining logic gates
The consistency of obtaining logic gates can be confirmedby estimating the probability of the desired output for dif-ferent noise strengths, where P(Logic) is estimated by calcu-lating the ratio of the number of runs which gives the cor-rect logic output to the total number of runs with differentinput streams. When the probability P(logic) is 1, the sys-tem reproduces completely reliable logic gates. This notionof probability would help to identify which region will ex-plicitly show the logic operation for different noise strengths0 < D < . E. Effect of bias and implementation of different logic gates
Now, we discuss how on changing the bias from positiveto negative value in the optimum range, the dynamics of thesystem gets varied. For this purpose we study the effect ofconstant bias ε in (1). From Fig.12, it is obvious that for theinput stream ( , ) the attractor is bounded in the x > x < ( , ) state andfor other input streams ( , ) / ( , ) it is again bounded in the FIG. 10. Poincaré section of the phase space with bias ε = . δ = . D = . D = . -5 -5 -5 -5 P ( l og i c ) D FIG. 11. Probability of obtaining logic output for various noisestrengths D with fixed forcing parameter F = . δ = . ε = . x > ε = .
1. Then the dy-namical attractor will produce logical OR gate [see Fig.12(a)].On the other hand, when we change the bias to ε = − . FIG. 12. Phase space plane for various values of ‘ ε ’. Panels (a)and (b) represent the logical OR gate with ε = . ε = − .
1, respectively, for fixed parameters F = . D = . δ = . -0.6 0 0.6 100000 110000 120000(a) I -0.1 0 0.1 100000 110000 120000(b) ε -1.001.0 100000 110000 120000(c) x ( t ) Time
FIG. 13. Panel (a) shows a combination of two input signals I = I + I . Input I + I = + . I + I = − . ε , whenit varies from 0 . − .
1. Panel (c) represents the correspondingdynamical response of the system x(t) under periodic forcing andnoise with fixed parameters F = .
44 and D = . ( . ∗ − . ∗ ) and ANDgate beyond 1 . ∗ time units. again, it is found that the response of the oscillator is in the x < x ( t ) > x ( t ) <
0. As a result, if we changethe bias from ε = . ε = − . I = I + I . Fig.13(b) represents the bias changing. In thisdiagram we observe that as the bias changes from positive tonegative optimum values the corresponding symmetry of the potential well alternates, which leads to a change from ORlogic gate to AND logic gate[Fig.13(c)]. V. CONCLUSION
In the present paper, we have studied the existence of noiseinduced SNAs in a single periodically driven Duffing oscilla-tor system. To test the validity of the robustness of such SNAs,we perturb the system by adding two logic signals which leadsto the emulation of different logical behaviors. For appropri-ate input signals and noise, we realized different logical out-puts in the above simple periodically driven nonlinear system.The present study is significant in the aspect that noise in-duced logical SNAs can be observed in periodically drivendouble-well Duffing oscillator with single periodic force . Uti-lizing this feature, we have explicitly demonstrated the imple-mentation of OR gate in the corresponding nonlinear system.On bias/threshold changing from positive to negative valuesthe logic gate switches over from OR to AND logic gate. Thuswe see that noise assisted periodically driven system can ex-hibit logic behavior via logical SNA. Noise-induced SNAs,logical SNAs and standard SNAs are characterized by finite-time Lyapunov exponents, spectral characteristics and snap-shot attractors to examine the strange and nonchaotic behav-iors.The logical stochastic resonance is realized only in an op-timal range of noise strength. That is, it cannot be observedfor low or high noise intervals. It is established that the struc-ture of the attractor gets smeared out by the large amplitudeof noise. Thus attractors in random dynamical systems arepurposeful only for small amplitude noise. Further, analogcomponents in the electronic circuits generate noise in the mi-crovolt range. Our study paves the way for implementationof logic elements only in the experimental regime of noisestrength. Further in chaotic computing, a small amount ofnoise can make the computing to drastically change. Thus,our study also demonstrates that the existence of logical ele-ments is possible in the SNA regime and that the phenomenonis robust in the presence of experimental noise. Thus noise in-duced SNA is a good candidate to realize reconfigurable, andflexible computer hardware.
AUTHOR’S CONTRIBUTIONS
All authors contributed equally.
ACKNOWLEDGMENT
M.S. sincerely thanks Council of Scientific & Industrial Re-search, India for providing a fellowship under SRF SchemeNo.08/711(0001)2K19-EMR-I. A.V. is supported by theDST-SERB research project Grant No.EMR/2017/002813.M.L. acknowledges the financial support under the DST-SERB Distinguished Fellowship program under GrantNo.SB/DF/04/2017.
DATA AVAILABILITY
The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.
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