Intermittency Route to Strange Nonchaotic Attractors

Prasad, Awadhesh; Mehra, Vishal; Ramaswamy, Ramakrishna

doi:10.1103/physrevlett.79.4127

Cited by 116 publications

(155 citation statements)

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“…Recently much attention has been paid to the study of quasiperiodically forced systems because of the generic appearance of strange nonchaotic attractors (SNAs) [1]. SNAs were first described by Grebogi et al [2] and since then have been extensively investigated both numerically [3,4,5,6,7,8,9,10,11,12,13,14,15,16] and experimentally [17]. SNAs exhibit some properties of regular as well as chaoic attractors.…”

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confidence: 99%

“…Therefore, dynamical transitions in quasiperiodically forced systems have become a topic of considerable current interest. However, their mechanisms are still much less understood than those in unforced or periodically forced systems.Here we are interested in the dynamical transition to SNAs accompanied by intermittent behavior, as reported in [12]. As a parameter passes a threshold value, a smooth torus is abruptly transformed into an intermittent SNA.…”

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Mechanism for the intermittent route to strange nonchaotic attractors

2003

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Intermittent strange nonchaotic attractors (SNAs) appear typically in quasiperiodically forced period-doubling systems. As a representative model, we consider the quasiperiodically forced logistic map and investigate the mechanism for the intermittent route to SNAs using rational approximations to the quasiperiodic forcing. It is thus found that a smooth torus is transformed into an intermittent SNA via a phase-dependent saddle-node bifurcation when it collides with a new type of "ring-shaped" unstable set. Besides this intermittent transition, other transitions such as the interior, boundary, and band-merging crises may also occur through collision with the ring-shaped unstable sets. Hence the ring-shaped unstable sets play a central role for such dynamical transitions. Furthermore, these kinds of dynamical transitions seem to be "universal," in the sense that they occur typically in a large class of quasiperiodically forced period-doubling systems. Recently much attention has been paid to the study of quasiperiodically forced systems because of the generic appearance of strange nonchaotic attractors (SNAs) [1]. SNAs were first described by Grebogi et al. [2] and since then have been extensively investigated both numerically [3,4,5,6,7,8,9,10,11,12,13,14,15,16] and experimentally [17]. SNAs exhibit some properties of regular as well as chaoic attractors. Like regular attractors, their dynamics is nonchaotic; like typical chaotic attractors, they exhibit fractal phase space structure. Furthermore, SNAs are related to Anderson localization in the Schrödinger equation with a spatially quasiperiodic potential [18], and they may have a practical application in secure communication [19]. Therefore, dynamical transitions in quasiperiodically forced systems have become a topic of considerable current interest. However, their mechanisms are still much less understood than those in unforced or periodically forced systems.Here we are interested in the dynamical transition to SNAs accompanied by intermittent behavior, as reported in [12]. As a parameter passes a threshold value, a smooth torus is abruptly transformed into an intermittent SNA. Near the transition point, the intermittent dynamics on the SNA was characterized in terms of the average interburst time and the Lyapunov exponent. This route to an intermittent SNA is quite general and has been observed in a number of quasiperiodically forced period-doubling maps and flows (e.g., see [13,14]). It has been suggested [15] that the observed intermittent behavior results through interaction with an unstable orbit. However, in the previous work, such an unstable orbit was not located, and thus the bifurcation mechanism for the intermittent transition still remains unclear.In this paper, we study the underlying mechanism of the intermittency in the quasiperiodically forced logistic map M [5] which is a representative model for the quasiperiodically forced period-doubling systems:M :where x ∈ [0, 1], θ ∈ S 1 , a is the nonlinearity parameter of the logistic map, and ω and ...

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Mechanism for the intermittent route to strange nonchaotic attractors

2003

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“…On employing a second harmonic perturbation to a periodically driven nonlinear system to exhibit the vibrational resonance phenomenon, it has been found [21][22][23][24][25][26][27][28][29][30] that if the frequency of the second harmonic force and the driving force are incommensurate, then their ratio will be irrational and one can find the occurrence of strange nonchaotic attractors (SNAs) in these systems. Such systems are designated as quasiperiodically driven nonlinear systems (QPDNS).…”

Section: Introductionmentioning

confidence: 99%

Design and implementation of dynamic logic gates and R-S flip-flop using quasiperiodically driven Murali-Lakshmanan-Chua circuit

Venkatesh¹,

Venkatesan²,

Lakshmanan

2017

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We report the propagation of a square wave signal in a quasi-periodically driven Murali-Lakshmanan-Chua (QPDMLC) circuit system. It is observed that signal propagation is possible only above a certain threshold strength of the square wave or digital signal and all the values above the threshold amplitude are termed as 'region of signal propagation'. Then, we extend this region of signal propagation to perform various logical operations like AND/NAND/OR/NOR and hence it is also designated as the 'region of logical operation'. Based on this region, we propose implementing the dynamic logic gates, namely AND/NAND/OR/NOR, which can be decided by the asymmetrical input square waves without altering the system parameters. Further, we show that a single QPDMLC system will produce simultaneously two outputs which are complementary to each other. As a result, a single QPDMLC system yields either AND as well as NAND or OR as well as NOR gates simultaneously. Then we combine the corresponding two QPDMLC systems in a cross-coupled way and report that its dynamics mimics that of fundamental R-S flip-flop circuit. All these phenomena have been explained with analytical solutions of the circuit equations characterizing the system and finally the results are compared with the corresponding numerical and experimental analysis. Nonlinear dynamics based computing is an emerging field which can replace silicon chips and is becoming increasingly popular in nonlinear and chaotic dynamics, and computing research. Actually, Boolean based silicon chips are extremely logical in their operations. These logical operations can be classified into two groups, namely combinational logic circuits and sequential logic circuits. The basic building blocks for combinational logic circuits are the logic gates, namely AND, OR, NOT, NAND, NOR, etc., whereas the basic building block for sequential logic circuits is the flip-flop. Here, we have proposed a new mechanism by using a quasi-periodically driven nonlinear dynamical system exhibiting a strange non-chaotic attractor for constructing the logic gates AND/NAND/OR/NOR and fundamental R-S flip-flop circuit with Murali-LakshmananChua circuit as an example. By only imposing constraints on the logic high and logic low values of the input signal, we are able to implement these dynamic logic gates and flip-flop. In fact, without altering the system parameters, we point out how the logical operations can be decided by the asymmetrical input square waves. Consequently, dynamic computing using the behaviour of nona) Electronic mail: venkatesh.sprv@gmail.com b) Electronic mail: av.phys@gmail.com c) Electronic mail: lakshman@cnld.bdu.ac.in linear dynamical systems is shown to be quite implementable in hardware devices. Our results also show that nonlinearity is more significant than the existence of chaos for design of the logic gates and latches.

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“…2 Thus far, SNAs have been studied in various nonlinear models [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and have been observed in experiments using electronic circuits, [22][23][24][25] magnetoelastic ribbons, 26 and electrochemical cells 27 (see also Refs. 28-30, and references therein).…”

Section: Introductionmentioning

confidence: 99%

Experimental distinction between chaotic and strange nonchaotic attractors on the basis of consistency

Uenohara

Mitsui

Hirata

et al. 2013

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We experimentally study strange nonchaotic attractors (SNAs) and chaotic attractors by using a nonlinear integrated circuit driven by a quasiperiodic input signal. An SNA is a geometrically strange attractor for which typical orbits have nonpositive Lyapunov exponents. It is a difficult problem to distinguish between SNAs and chaotic attractors experimentally. If a system has an SNA as a unique attractor, the system produces an identical response to a repeated quasiperiodic signal, regardless of the initial conditions, after a certain transient time. Such reproducibility of response outputs is called consistency. On the other hand, if the attractor is chaotic, the consistency is low owing to the sensitive dependence on initial conditions. In this paper, we analyze the experimental data for distinguishing between SNAs and chaotic attractors on the basis of the consistency. Strange nonchaotic attractors (SNAs) often appear in dynamical systems driven by a quasiperiodic signal. An SNA is a geometrically strange attractor for which typical orbits have nonpositive Lyapunov exponents. In fact, typical orbits are insensitive to initial conditions under a common quasiperiodic signal. Owing to their characteristics reminiscent of both quasiperiodic order and chaos, SNAs have attracted the attention of researchers from both theoretical and experimental perspectives. Conventionally, the information dimension is often employed to distinguish between SNAs and chaotic attractors in experiments. However, using the information dimension for the experimental distinction between these types of attractors is difficult as the estimation of the information dimension is highly sensitive to noise, and requires rather long time series. Recently, Ngamga et al. 1 proposed a method for distinguishing between SNAs and chaotic attractors, which utilizes the consistency property of SNAs. Consistency here means that a nonlinear system shows the same response to the same input signal after some transient time, regardless of the initial conditions. In the method proposed by Ngamga et al., the determinism for a cross-recurrence plot of two response time series to the same input signal is used as a consistency measure. However, to our knowledge, there is no research paper on using their cross-recurrence method to experimentally distinguish between SNAs and chaotic attractors. In this study, we evaluated the consistency of a system by using the zero-delay normalized cross-correlation and Ngamga et al.'s cross-recurrence methods. By combining spectrum analysis and evaluation of consistency, we showed that SNAs and chaotic attractors can be distinguished more precisely as compared to conventional methods.

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Intermittency Route to Strange Nonchaotic Attractors

Cited by 116 publications

References 24 publications

Mechanism for the intermittent route to strange nonchaotic attractors

Mechanism for the intermittent route to strange nonchaotic attractors

Design and implementation of dynamic logic gates and R-S flip-flop using quasiperiodically driven Murali-Lakshmanan-Chua circuit

Experimental distinction between chaotic and strange nonchaotic attractors on the basis of consistency

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