^{1,2}, A. Venkatesan

^{1}and M. Lakshmanan

^{2}

### Abstract

We show that the recently introduced 0-1 test can successfully distinguish between strange nonchaotic attractors (SNAs) and periodic/quasiperiodic/chaotic attractors, by suitably choosing the arbitrary parameter associated with the translation variables in terms of the golden mean number which avoids resonance with the quasiperiodic force. We further characterize the transition from quasiperiodic to chaotic motion via SNAs in terms of the 0-1 test. We demonstrate that the test helps to detect different dynamical transitions to SNAs from quasiperiodic attractor or the transitions from SNAs to chaos. We illustrate the performance of the 0-1 test in detecting transitions to SNAs in quasiperiodically forced logistic map, cubic map, and Duffing oscillator.

A strange nonchaotic attractor (SNA) is considered as a complicated structure in phase space. Such a complex structure is a typical property usually associated with a chaotic attractor. However, SNAs are nonchaotic in a dynamical sense because they do not exhibit sensitive dependence on initial conditions (as evidenced by negative Lyapunov exponents). These attractors are ubiquitous in different quasiperiodically driven nonlinear systems. While the existence of SNAs has been firmly established, a question of intense interest is how do these attractors morphologically differ from other attractors and how they can be characterized and quantified. Two specific characterizations, namely, finite time Lyapunov exponents (FTLEs) and recurrence plots were proposed earlier in the literature. However, in the present work we show that the recently introduced measure of 0-1 test clearly distinguishes quasiperiodic motion, SNAs, and chaotic attractors and also allows one to detect different dynamical transitions from quasiperiodic motion to SNAs and SNAs to chaotic attractors.

The work of R.G. and M.L. has been supported by the Department of Science and Technology (DST), Government of India sponsored IRHPA research project. M.L. was also supported by a Department of Atomic Energy Raja Ramanna fellowship and a DST Ramanna program. A.V. acknowledges the support from UGC minor research project.

I. INTRODUCTION

II. UNDERSTANDING OF 0-1 TEST

A. Translation variables and the 0-1 test for regular and chaotic motion

B. Extending the 0-1 test to quasiperiodically forced systems exhibiting SNAs

III. APPLICABILITY OF 0-1 TEST IN QUASIPERIODICALLY DRIVEN SYSTEMS

A. Dynamics in the absence of quasiperiodic force

B. Dynamics with quasiperiodic force

IV. DIFFERENT ROUTES TO SNAs

A. Heagy-Hammel route

B. Fractalization route

C. Intermittency route

V. THE TRANSITIONS FROM SNAs TO CHAOTIC ATTRACTORS

VI. CONCLUSION

### Key Topics

- Attractors
- 52.0
- Chaos
- 23.0
- Time series analysis
- 16.0
- Chaotic dynamics
- 15.0
- Oscillators
- 13.0

## Figures

Plots of (i) translation variable p versus q, (ii) mean square displacement versus n 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 N = 5 × 104 with n = N/10.

Plots of (i) translation variable p versus q, (ii) mean square displacement versus n 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 N = 5 × 104 with n = N/10.

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 (K) from 0-1 test as a function of system parameter. Here N = 5 × 104.

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 (p, q) in terms of 0-1 test (Eqs. (1) and (2) ); (iii) mean square displacement as a function of n. Here N = 5 × 104 with n = N/10.

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 (p, q) in terms of 0-1 test (Eqs. (1) and (2) ); (iii) mean square displacement as a function of n. Here N = 5 × 104 with n = N/10.

(a) Variation of the largest Lyapunov exponent (b) variation of the asymptotic growth rate (K) 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, N = 3 × 105 with L = 104 as the length of each segment.

(a) Variation of the largest Lyapunov exponent (b) variation of the asymptotic growth rate (K) 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, N = 3 × 105 with L = 104 as the length of each segment.

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 (K) from 0-1 test. Here, N = 3 × 105 with L = 104.

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 (K) from 0-1 test. Here, N = 3 × 105 with L = 104.

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 (K) from 0-1 test. Here, N = 5 × 105 (after sampling the data points) and L = 104.

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 (K) from 0-1 test. Here, N = 5 × 105 (after sampling the data points) and L = 104.

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 (K) from 0-1 test. Here, N = 3 × 105 with L = 104 as the length of each segment.

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 (K) from 0-1 test. Here, N = 3 × 105 with L = 104 as the length of each segment.

Transition from SNA to torus. (a) Behavior of the Lyapunov exponent (λ). (b) Behavior of the asymptotic growth rate (K) from 0-1 test during the fractalization route in the parametrically driven double well Duffing oscillator. Here 5 × 105 (after sampling the data points) with L = 104.

Transition from SNA to torus. (a) Behavior of the Lyapunov exponent (λ). (b) Behavior of the asymptotic growth rate (K) from 0-1 test during the fractalization route in the parametrically driven double well Duffing oscillator. Here 5 × 105 (after sampling the data points) with L = 104.

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 (K) from 0-1 test. Here N = 3 × 105 with L = 104.

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 (K) from 0-1 test. Here N = 3 × 105 with L = 104.

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 (K) from 0-1 test. Here, N = 5 × 105 (after sampling the data points) with L = 104 as the length of each segment.

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 (K) from 0-1 test. Here, N = 5 × 105 (after sampling the data points) with L = 104 as the length of each segment.

(a) Variation of the Lyapunov exponent λ and (b) variation of the asymptotic growth rate (K) from 0-1 test as a function of α indicating transition from SNAs to chaotic attractors in the quasiperiodically forced logistic map with . Here N = 3 × 105 and L = 104.

(a) Variation of the Lyapunov exponent λ and (b) variation of the asymptotic growth rate (K) from 0-1 test as a function of α indicating transition from SNAs to chaotic attractors in the quasiperiodically forced logistic map with . Here N = 3 × 105 and L = 104.

(a) Variation of the Lyapunov exponent λ and (b) variation of the asymptotic growth rate K as a function of H indicating transition from chaos to SNAs in the parametrically driven double well Duffing oscillator with R = 0.47. Here N = 5 × 105 (after sampling the data points) with L = 104.

(a) Variation of the Lyapunov exponent λ and (b) variation of the asymptotic growth rate K as a function of H indicating transition from chaos to SNAs in the parametrically driven double well Duffing oscillator with R = 0.47. Here N = 5 × 105 (after sampling the data points) with L = 104.

Choosing the most optimal value of c: Attractors of the quasiperiodically forced logistic map in the translation variables (p, q) space. (i) torus, (ii) SNA, and (iii) chaotic attractors for the following choices of c: (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 N = 5 × 104.

Choosing the most optimal value of c: Attractors of the quasiperiodically forced logistic map in the translation variables (p, q) space. (i) torus, (ii) SNA, and (iii) chaotic attractors for the following choices of c: (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 N = 5 × 104.

Nature of the translation variables (p, q) for the quasiperiodically forced logistic map (Eq. (11) ) as a function of the length of the time series N for (a) torus for , (b) SNA for , (c) chaotic attractor for : (i) N = 5 × 104, (ii) N = 1 × 105, (iii) N = 2 × 105, (iv) N = 3 × 105.

Nature of the translation variables (p, q) for the quasiperiodically forced logistic map (Eq. (11) ) as a function of the length of the time series N for (a) torus for , (b) SNA for , (c) chaotic attractor for : (i) N = 5 × 104, (ii) N = 1 × 105, (iii) N = 2 × 105, (iv) N = 3 × 105.

Plots of (a) asymptotic growth rate K. (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 (C) for .

Plots of (a) asymptotic growth rate K. (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 (C) for .

For the quasiperiodically forced cubic map (Eq. (12) ). (a) Torus for f = 0.10, A = 2.14, (b) SNA for f = 0.7, A = 1.86687, and (c) chaotic attractor for f = 0.10, A = 2.19: (i) projection of the attractor; (ii) dynamics of the translation variables (p, q) in terms of 0-1 test (Eqs. (1) and (2) ); (iii) mean square displacement M(n) as a function of n. Here N = 5 × 104 with n = N/10.

For the quasiperiodically forced cubic map (Eq. (12) ). (a) Torus for f = 0.10, A = 2.14, (b) SNA for f = 0.7, A = 1.86687, and (c) chaotic attractor for f = 0.10, A = 2.19: (i) projection of the attractor; (ii) dynamics of the translation variables (p, q) in terms of 0-1 test (Eqs. (1) and (2) ); (iii) mean square displacement M(n) as a function of n. Here N = 5 × 104 with n = N/10.

A comparative distribution of asymptotic growth rate P(K, N) as a function of K 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.

A comparative distribution of asymptotic growth rate P(K, N) as a function of K 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.

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 f = 0.7, and A = 1.8865 (torus) and A = 1.8870 (SNA); (c), (d) Fractalization route with f = 0.1, and A = 2.14 (torus) and A = 2.167 (SNA); (e), (f) intermittency route with f = 0.7, A = 1.8017 (torus), and A = 1.801685 (SNA).

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 f = 0.7, and A = 1.8865 (torus) and A = 1.8870 (SNA); (c), (d) Fractalization route with f = 0.1, and A = 2.14 (torus) and A = 2.167 (SNA); (e), (f) intermittency route with f = 0.7, A = 1.8017 (torus), and A = 1.801685 (SNA).

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