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Applicability of 0-1 test for strange nonchaotic attractors
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10.1063/1.4808254
/content/aip/journal/chaos/23/2/10.1063/1.4808254
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/2/10.1063/1.4808254
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Plots of (i) translation variable versus , (ii) mean square displacement versus for the logistic map (Eq. (11) ) corresponding to (a) periodic dynamics for and (b) chaotic dynamics for . Here, the total length of the time series  = 5 × 10 with  = /10.

Image of FIG. 2.
FIG. 2.

Application of 0-1 test for the force-free case of (a) logistic map (Eq. (11) ) and (b) cubic map (Eq. (12) ): (i) bifurcation diagram, (ii) asymptotic growth rate () from 0-1 test as a function of system parameter. Here  = 5 × 10.

Image of FIG. 3.
FIG. 3.

For the quasiperiodically forced logistic map. (a) Torus motion for , (b) SNA for , and (c) chaotic attractor for : (i) projection of the attractor; (ii) dynamics of the translation components (, ) in terms of 0-1 test (Eqs. (1) and (2) ); (iii) mean square displacement as a function of . Here  = 5 × 10 with  = /10.

Image of FIG. 4.
FIG. 4.

(a) Variation of the largest Lyapunov exponent (b) variation of the asymptotic growth rate () from 0-1 test as a function of α for fixed indicating torus → SNA → chaos transitions in the quasiperiodically forced logistic map. The vertical arrows in this and the following figures indicate transition points. Here,  = 3 × 10 with  = 10 as the length of each segment.

Image of FIG. 5.
FIG. 5.

Transition from torus to SNA through HH mechanism in the (a) logistic map and (b) cubic map: (i) behavior of the Lyapunov exponent (λ); (ii) behavior of the asymptotic growth rate () from 0-1 test. Here,  = 3 × 10 with  = 10.

Image of FIG. 6.
FIG. 6.

Transition from SNA to torus through HH mechanism in the Duffing oscillator. (a) Behavior of the Lyapunov exponent (λ). (b) Behavior of the asymptotic growth rate () from 0-1 test. Here,  = 5 × 10 (after sampling the data points) and  = 10.

Image of FIG. 7.
FIG. 7.

Transition from torus to SNA during fractalization route in the (a) logistic map (b) cubic map: (i) behavior of Lyapunov exponent (λ) and (ii) behavior of asymptotic growth rate () from 0-1 test. Here,  = 3 × 10 with  = 10 as the length of each segment.

Image of FIG. 8.
FIG. 8.

Transition from SNA to torus. (a) Behavior of the Lyapunov exponent (λ). (b) Behavior of the asymptotic growth rate () from 0-1 test during the fractalization route in the parametrically driven double well Duffing oscillator. Here 5 × 10 (after sampling the data points) with  = 10.

Image of FIG. 9.
FIG. 9.

Transition from a SNA to a torus through type I intermittency in the (a) logistic map and (b) cubic map: (i) behavior of the Lyapunov exponent (λ); (ii) behavior of the asymptotic growth rate () from 0-1 test. Here  = 3 × 10 with  = 10.

Image of FIG. 10.
FIG. 10.

Transition from intermittent SNA to torus motion in the two frequency parametrically driven double well Duffing oscillator: (a) behavior of the Lyapunov exponent (λ). (b) Behavior of the asymptotic growth rate () from 0-1 test. Here,  = 5 × 10 (after sampling the data points) with  = 10 as the length of each segment.

Image of FIG. 11.
FIG. 11.

(a) Variation of the Lyapunov exponent λ and (b) variation of the asymptotic growth rate () from 0-1 test as a function of α indicating transition from SNAs to chaotic attractors in the quasiperiodically forced logistic map with . Here  = 3 × 10 and  = 10.

Image of FIG. 12.
FIG. 12.

(a) Variation of the Lyapunov exponent λ and (b) variation of the asymptotic growth rate as a function of indicating transition from chaos to SNAs in the parametrically driven double well Duffing oscillator with R = 0.47. Here  = 5 × 10 (after sampling the data points) with  = 10.

Image of FIG. 13.
FIG. 13.

Choosing the most optimal value of : Attractors of the quasiperiodically forced logistic map in the translation variables (, ) space. (i) torus, (ii) SNA, and (iii) chaotic attractors for the following choices of : (a) , (b) , (c) , (d) , (e) , (f) , where and . Note that the choice (e) with gives the most satisfactory description of all the three attractors. Here  = 5 × 10.

Image of FIG. 14.
FIG. 14.

Nature of the translation variables (, ) for the quasiperiodically forced logistic map (Eq. (11) ) as a function of the length of the time series for (a) torus for , (b) SNA for , (c) chaotic attractor for : (i)  = 5 × 10, (ii)  = 1 × 10, (iii)  = 2 × 10, (iv)  = 3 × 10.

Image of FIG. 15.
FIG. 15.

Plots of (a) asymptotic growth rate . (b) Lyapunov exponent as a function of time series in the quasiperiodically forced logistic map (Eq. (11) ). Here, T indicates torus for ; SNA for ; chaotic attractor () for .

Image of FIG. 16.
FIG. 16.

For the quasiperiodically forced cubic map (Eq. (12) ). (a) Torus for  = 0.10,  = 2.14, (b) SNA for  = 0.7,  = 1.86687, and (c) chaotic attractor for  = 0.10,  = 2.19: (i) projection of the attractor; (ii) dynamics of the translation variables (, ) in terms of 0-1 test (Eqs. (1) and (2) ); (iii) mean square displacement () as a function of n. Here  = 5 × 10 with  = /10.

Image of FIG. 17.
FIG. 17.

A comparative distribution of asymptotic growth rate (, ) as a function of for the three transitions showing its values for a torus (Δ) just before transition and a SNA (continuous curve) just after transition in the case of the quasiperiodically driven logistic map (Eq. (11) ): (a) HH route for the parameter (for torus) and (for SNA); (c) Fractalization route (for torus) and (SNA); (e) intermittency route with and (torus), and (SNA). The corresponding distribution of as a function of λ are given in (b), (d), and (f), respectively, for the above three routes for comparison purpose.

Image of FIG. 18.
FIG. 18.

Same as Fig. 17 but now for the case of quasiperiodically forced cubic map (Eq. (12) ). The choice of parameters here are as follows. (a), (b) HH route with  = 0.7, and  = 1.8865 (torus) and  = 1.8870 (SNA); (c), (d) Fractalization route with  = 0.1, and  = 2.14 (torus) and  = 2.167 (SNA); (e), (f) intermittency route with  = 0.7,  = 1.8017 (torus), and  = 1.801685 (SNA).

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/content/aip/journal/chaos/23/2/10.1063/1.4808254
2013-06-05
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Applicability of 0-1 test for strange nonchaotic attractors
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/2/10.1063/1.4808254
10.1063/1.4808254
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