# Characterizing Complex Dynamics in the Classical and Semi-Classical Duffing Oscillator Using Ordinal Patterns Analysis

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## Abstract

**:**

## 1. Introduction

## 2. The Duffing Oscillator

## 3. Ordinal Patterns Analysis and Permutation Entropy

## 4. Lyapunov Exponent and Poincaré Sections

## 5. Results and Discussion

#### 5.1. The Lyapunov Exponent

#### 5.2. Ordinal Pattern Analysis of the Classical Duffing Oscillator

#### 5.3. Ordinal Pattern Analysis of the Semi-Classical Duffing Oscillator

#### 5.4. Permutation Entropy and Relation with Lyapunov Exponent

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Position of the oscillator versus time. The green squares indicate the detection of a peak, and the red dots indicate the crossing from one well to the other. Indicated with black arrows is depicted one word of dimension D = 3 as example. The word is computed with inter-peak intervals. Because $TI(i)<TI(i+1)<TI(i+2)$, we designate the word as “012”. The orange arrows indicate three consecutive inter-crossing intervals. These time-intervals are equal, indicating certain regularity in the dynamics for crossing from one well to the other.

**Figure 2.**Time series of the dynamical system for various values of the damping parameter, $\mathsf{\Gamma}$. The green dots indicate the detection of a peak, and the red dots indicate the crossing from one well to the other.

**Figure 3.**Poincaré section for the various values of the damping parameter in Figure 2. Horizontal axis corresponds to position, x, and vertical axis to momentum, p. $\mathsf{\Gamma}=0.06$ and $\mathsf{\Gamma}=0.23$ reflect the periodic behavior while for $0.07<\mathsf{\Gamma}<0.21$ the system is chaotic. For $\mathsf{\Gamma}=0.21$ this recurrence map shows how the system only moves in one of the two wells. Poincaré sections of the system do not indicate any transition in the chaotic regime. The attractor evolves smoothly as the control parameter, $\mathsf{\Gamma}$, is changed.

**Figure 4.**(

**a**,

**b**) Lyapunov exponent versus the damping parameter, $\mathsf{\Gamma}$; (

**c**,

**d**) Words probabilities of dimension 3 considering the peaks of the time series as events. Six regions can be differentiated from the distinct hierarchies of the words; (

**e**,

**f**) Words probabilities of the dynamics considering as events the crossings from one potential well to the other potential well. The gray dotted line corresponds to the words that contain one, two or three equal consecutive time intervals. Left row corresponds to the classical Duffing oscillator while right row to the semi-classical.

**Figure 5.**Words probabilities computed with the peaks (

**a**); and with the crossings (

**b**), for region I. The dotted line corresponds to the word with consecutive equal time intervals. The probability for the periodic word is much higher than for the rest of the words.

**Figure 6.**(

**a**) Lyapunov exponent versus the damping parameter, $\mathsf{\Gamma}$; (

**b**) Permutation entropy (PE) computed with words of dimension 3 with the peaks; (

**c**) PE computed with words of dimension 3 with the crossings.

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**MDPI and ACS Style**

Trostel, M.L.; Misplon, M.Z.R.; Aragoneses, A.; Pattanayak, A.K.
Characterizing Complex Dynamics in the Classical and Semi-Classical Duffing Oscillator Using Ordinal Patterns Analysis. *Entropy* **2018**, *20*, 40.
https://doi.org/10.3390/e20010040

**AMA Style**

Trostel ML, Misplon MZR, Aragoneses A, Pattanayak AK.
Characterizing Complex Dynamics in the Classical and Semi-Classical Duffing Oscillator Using Ordinal Patterns Analysis. *Entropy*. 2018; 20(1):40.
https://doi.org/10.3390/e20010040

**Chicago/Turabian Style**

Trostel, Max L., Moses Z. R. Misplon, Andrés Aragoneses, and Arjendu K. Pattanayak.
2018. "Characterizing Complex Dynamics in the Classical and Semi-Classical Duffing Oscillator Using Ordinal Patterns Analysis" *Entropy* 20, no. 1: 40.
https://doi.org/10.3390/e20010040