# Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems

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## Abstract

**:**

## 1. Introduction

## 2. Lyapunov Exponents

#### 2.1. The Largest Lyapunov Exponent

_{1}and x

_{2}were chosen (in the phase space). They stand for the origins of the trajectories $({x}_{1}\left(t\right)$ and ${x}_{2}\left(t\right))$. The change in the distance d between two corresponding points of these trajectories under evolution of system (1) can be monitored by:

#### 2.2. Results

- (1)
- n = 1. In this case only a stable fixed point can be an attractor (node or focus). There exists one negative LE denoted by ${\lambda}_{\text{}1}=(-),$
- (2)
- n = 2. In 2D systems, there are two types of attractors: stable fixed points and limit cycles. The corresponding LEs follow:
- $({\lambda}_{1},{\lambda}_{2})=(-,-)$—stable fixed/fixed point;
- $({\lambda}_{1},{\lambda}_{2})=(0,-)$—stable limit cycle (one exponent is equal to zero).

- (3)
- n = 3. In 3D phase space, there exist four types of attractors: stable points, limit cycles, 2D tori and strange attractors. The following set of LEs characterizes possible dynamical situations to be met:
- $({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})=(-,-,-)$—stable fixed point;
- $({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})=(0,-,-)$—stable limit cycle;
- $({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})=(0,0,-)$—stable 2D tori;
- $({\lambda}_{1},{\lambda}_{2},{\lambda}_{3})=(+,0,-)$—strange attractor.

## 3. Methods of Analysis of Lyapunov Exponents

#### 3.1. Benettin Method

#### 3.2. Wolf Method

#### 3.3. Rosenstein Method

#### 3.4. Kantz Method

#### 3.5. Computation of LLE Based on Synchronization of Nonnegative Feedback

#### 3.6. Jacobi Method

#### 3.7. Modification of the Neural Network Method

- (i)
- the network is sensitive to the input information (information is given in the form of real numbers);
- (ii)
- the network is self-organizing, i.e., it yields the output space of solutions only based on the inputs;
- (iii)
- the neural network is a network of straight distribution (all connections are directed from input neurons to output neurons);
- (iv)
- owing to the synapses tuning, the network exhibits dynamic couplings (in the learning process, the tuning of the synaptic coupling takes place $(dW/dt\ne 0),$ where W stands for the weighted coefficients of the network).

## 4. Wavelet Methods

#### Gauss Wavelets

## 5. Analysis of Classical Dynamical Systems by LEs and Gauss Wavelets

#### 5.1. Logistic Map

_{1}= 0.693147181, and the Kaplan–Yorke dimension: 1.0.

#### 5.2. Hénon Map

#### 5.3. Hyperchaotic Generalised Hénon Map

#### 5.4. Rössler Attractor

#### 5.5. Lorenz Attractor

## 6. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Synchronization of perturbed and nonperturbed systems in the case of a logistic map ($\lambda $ points to the largest Lyapunov exponent value).

**Figure 2.**Transformation of a sphere of initial states into a counterpart ellipsoid during the system evolution.

**Figure 3.**Single-layer feed forward neural network, which consists of input neurons, a layer of hidden neurons and one output neuron.

**Figure 5.**Nonlinear

**c**haracteristics of the oscillation signal: (

**a**) Time histories; (

**b**) Time window; (

**c**) Chaotic attractor; (

**d**) Fourier frequency spectrum; (

**e**) Wavelet spectrum; (

**f**) Dependence of LLE on the control parameter.

**Figure 6.**Characteristics of the Hénon map: (

**a**) Time history; (

**b**) Time window; (

**c**) Chaotic attractor; (

**d**) Fourier frequency spectrum; (

**e**) Wavelet spectrum; (

**f**) Dependence of LLE on the control parameter; (

**g**) Lyapunov exponents plane (Hénon map).

**Figure 7.**Signal characteristics: (

**a**) Time history; (

**b**) Time window; (

**c**) Chaotic attractor; (

**d**) Fourier frequency spectrum; (

**e**) Wavelet spectrum; (

**f**) Dependence of LLE on the control parameter; (

**g**) Lyapunov exponents plane (generalized Hénon map).

**Figure 8.**Signal characteristics: (

**a**) Time history; (

**b**) Time window; (

**c**) Chaotic attractor; (

**d**) Fourier frequency spectrum; (

**e**) Wavelet spectrum; (

**f**) Dependence of LLE on the control parameter; (

**g**) Lyapunov exponents plane (Rössler attractor).

**Figure 9.**Signal characteristics: (

**a**) Time history; (

**b**) Time window; (

**c**) Chaotic attractor; (

**d**) Fourier frequency spectrum; (

**e**) Wavelet spectrum; (

**f**) Dependence of LLE on the control parameter; (

**g**) Lyapunov exponents plane (Lorenz attractor).

LE Spectrum | |||

Benettin Method | Neural Network | ||

(LEs): 0.69315 Dimension Kaplan–Yorke (DKY): 1 Kolmogorov-Sinai entropy (KSE): 0.69315 Phase volume compression (PVC): 0.69315 | LEs: 0.69290 DKY: 1 EKS: 0.69290 PVC: 0.69290 | ||

LLE | |||

Wolf Method | Rosenstein Method | Kantz Method | Method of Synchronization |

LLE: 0.99683 | LLE: 0.690553 | LLE: 0.69810 | LLE: 0.696 |

Spectrum of LLEs | |||

Benettin Method | Neural Network | ||

LEs: 0.41919; −1.62316 DKY: 1.25826 EKS: 0.41919 PVC: −1.20397 | LEs: 0.41919; −1.62316 DKY: 1.25826 EKS: 0.41919 PVC: −1.20397 | ||

LLEs | |||

Wolf Method | Rosenstein Method | Kantz Method | Synchronization Method |

LLE: 0.38788 | LLE: 0.414218 | LLE: 0.41912 | LLE: 0.40608 |

**Table 3.**Lyapunov exponents spectrum and LLEs computed by different methods (generalized Hénon map).

Spectrum of LEs | |||

Benettin Method | Neural Network | ||

LEs: 0.27628; 0.25770; −4.04053 DKY: 2.13215 EKS: 0.53397 PVC: −3.50656 | LEs: 0.29251; 0.27104; −4.04583 DKY: 2.13929 EKS: 0.56355 PVC: −3.48227 | ||

LLEs | |||

Wolf Method | Rosenstein Method | Kantz Method | Synchronization Method |

LLE: 0.45214 | LLE: 0.27930 | LLE: 0.26601 | 0.27250 |

Spectrum of LEs | ||

Benettin Method | Neural Network | |

LE: 0.07135; 0.00000; −5.39420 DKY: 2.01323 KSE: 0.07135 PVC: −5.32285 | LE: 0.07593; −0.00060; −0.78178 DKY: 2.09635 EKS: 0.07593 PVC: −0.70646 | |

LLEs | ||

Wolf Method | Rosenstein Method | Kantz Method |

LLE: 0.05855 | LLE: 0.0726 | LLE: 0.0774 |

Spectrum of LEs | ||

Benettin Method | Neural Network Method | |

LE: 0.90557; 0.00000; −14.57214 DKY: 2.06214 EKS: 0.90557 PVC: −13.66656 | LE: 0.9490; 0.0610; −13.9101 DKY: 2.07261 EKS: 1.0101 PVC: −12.9000 | |

LLEs | ||

Wolf Method | Rosenstein Methhod | Kantz Method |

LLE: 0.81704 | LLE: 0.836 | LLE: 0.807185 |

**Table 6.**Fourier power spectra (

**a**) and Gauss wavelet spectra (

**b**) obtained for $\Delta t=1,2$ and the LLEs computed by different methods (logistic map).

$\Delta \mathit{t}=\mathbf{1}$ | $\Delta \mathit{t}=\mathbf{2}$ |

Fourier Power Spectra (a) | |

Gauss Wavelet Spectra (b) | |

LLE (Wolf) | |

0.99961 | 1.00014 |

LLE (Rosenstein) | |

0.69231 | 0.69065 |

LLE (Kantz) | |

0.6981 | 0.69005 |

LLE (Synchronization) | |

0.69400 | 0.69330 |

LEs (Benettin) | |

LES: 0.69318 DKY: 1.00000 KSE: 0.69318 PVC: 0.69318 | LES: 0.69400 DKY: 1.00000 KSE: 0.69400 PVC: 0.69400 |

LEs (Neural Network) | |

LES: 0.69290 DKY: 1 SE: 0.69290 PVC: 0.69290 | LES: 0.69107 DKY: 1.00000 KSE: 0.69107 PVC: 0.69107 |

**Table 7.**Fourier power spectra (

**a**) and Gauss wavelet spectra (

**b**) obtained for $\Delta t=1,2$ and the computed LLEs by different methods (Hénon map).

$\Delta \mathit{t}=\mathbf{1}$ | $\Delta \mathit{t}=\mathbf{2}$ |

Fourier Power Spectra (a) | |

Gauss Wavelet Spectra (b) | |

LLE (Wolf) | |

0.4158 | 0.39734 |

LLE (Rosenstein) | |

0.41637 | 0.400635 |

LLE (Kantz) | |

0.41912 | 0.41478 |

LLE (Synchronization) | |

0.40608 | 0.40510 |

All LEs (Benettin) | |

LEs: 0.41919; −1.62316 DKY: 1.25826 EKs: 0.41919 PVC: −1.20397 | LEs: 0.41917; −1.62315 DKY: 1.25825 EKs: 0.41917 PVC: −1.20397 |

All LEs (Neural Network) | |

LEs: 0.41919; −1.62316 DKY: 1.25826 KSE: 0.41919 PVC: −1.20397 | LEs: 0.40924; −1.61321 DKY: 1.25368 KSE: 0.40924 PVC: −1.20397 |

**Table 8.**Fourier power spectra (

**a**) and Gauss wavelet spectra (

**b**) obtained for $\Delta t=1,2$ and the computed LLEs by different methods (generalized Hénon map).

$\Delta \mathit{t}=\mathbf{1}$ | $\Delta \mathit{t}=\mathbf{2}$ |

Fourier Power Spectra (a) | |

Gauss Wavelet Spectra (b) | |

LLE (Wolf) | |

0.45214 | 0.46706 |

LLE (Rosenstein) | |

0.27930 | 0.27459 (0.62515) |

LLE (Kantz) | |

0.26601 | 0.3359 |

LLE (Synchronization) | |

0.27250 | 0.27200 |

All LEs (Benettin) | |

LEs: 0.27628; 0.25770; −4.04053 DKY: 2.13215 KSE: 0.53397 PVC: −3.50656 | LEs: 0.27487; 0.25631; −4.03774 DKY: 2.13155 EKS: 0.53118 PVC: −3.50656 |

All LEs (Neural Network) | |

LEs: 0.29251; 0.27104; −4.04583 DKY: 2.13929 KSE: 0.56355 PVC: −3.48227 | LEs: 0.26304; 0.24387; −4.14321 DKY: 2.12235 KSE: 0.50691 PVC: −3.63630 |

**Table 9.**Fourier power spectra and Gauss wavelet spectra obtained for $\Delta t=0.05,0.1,0.15,0.2$ and the computed LLEs by different methods

**(**Rössler attractor).

$\Delta \mathit{t}=\mathbf{0.05}$ | $\Delta \mathit{t}=\mathbf{0.1}$ | $\Delta \mathit{t}=\mathbf{0.15}$ | $\Delta \mathit{t}=\mathbf{0.2}$ |

Fourier Power Spectrum | |||

Gauss Wavelets | |||

LLE (Wolf) | |||

0.07283 | 0.05855 | 0.01731 | 0.02544 |

LLE (Rosenstein) | |||

0.083 | 0.0726 | 0.06553 | 0.606 |

LLE (Kantz) | |||

0.0234 | 0.0208 | 0.02133 | 0.0215 |

All LEs (Benettin) | |||

LES: 0.07156; 0.00000; −5.38768 DKY: 2.01328 KSE: 0.07156 PVC: −5.31612 | LES: 0.06959; 0.00000; −5.21949 DKY: 2.01333 KSE: 0.06959 PVC: −5.14990 | LES: 0.06789; 0.00000; −4.34385 DKY: 2.01563 KSE: 0.06789 PVC: −4.27596 | LES: 0.06205; −0.00001; −2.84111 DKY: 2.02184 KSE: 0.06205 PVC: −2.77906 |

All LEs (neural network) | |||

LES: 0.06259; −0.07984; −0.32528 DKY: 1.78396 KSE: 0.06259 PVC: −0.34253 | LES: 0.07340; −0.02681; −0.23525 DKY: 2.19807 KSE: 0.07340 PVC: −0.18865 | LES: 0.07374; 0.00057; −0.36909 DKY: 2.20135 KSE: 0.07432 PVC: −0.29477 | LES: 0.07983; −0.02816; −0.91182 DKY: 2.05667 KSE: 0.07983 PVC: −0.86015 |

**Table 10.**Fourier power spectra and Gauss wavelet spectra obtained for $\Delta t=0.005,0.01,0.015,0.02$ and the computed LLEs by different methods (Lorenz attractor).

$\Delta \mathit{t}=\mathbf{0.005}$ | $\Delta \mathit{t}=\mathbf{0.01}$ | $\Delta \mathit{t}=\mathbf{0.015}$ | $\Delta \mathit{t}=\mathbf{0.02}$ |

Fourier Power Spectrum | |||

Gauss Wavelet | |||

LLE (Wolf) | |||

0.9721 | 0.81704 | 0.867 | 0.712 |

LLE (Rosenstein) | |||

0.876 | 0.836 | 0.858 | 0.859 |

LLE (Kantz) | |||

0.898 | 0.9 | 0.762667 | 0.84 |

LES (Benettin) | |||

LES: 0.90632; 0.00000; −14.57297 DKY: 2.06219 KSE: 0.90632 PVC: −13.66666 | LES: 0.90523; 0.00000; −14.57179 DKY: 2.06212 KSE: 0.90523 PVC: −13.66656 | LES: 0.90551; 0.00000; −14.57163 DKY: 2.06214 KSE: 0.90551 PVC: −13.66613 | LES: 0.90596; 0.00000; −14.57086 DKY: 2.06218 KSE: 0.90596 PVC: −13.66490 |

LES (Neural Network) | |||

LES: 0.91677; 0.04404; −6.464 DKY: 2.14864 EKS entropy: 0.96081 PVC: 0.89617 | LE: 0.9490; 0.0610; −13.9101 DKY: 2.07261 EKS: 1.0101 PVC: −12.9000 | LES: 0.8913; −0.3508; −14.3577 DKY: 2.03765 EKS: 0.8913 PVC: −13.8172 | LES: 0.7485; −0.05558; −23.3505 DKY: 2.0296 EKS: 0.7485 PVC: −22.65758 |

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**MDPI and ACS Style**

Awrejcewicz, J.; Krysko, A.V.; Erofeev, N.P.; Dobriyan, V.; Barulina, M.A.; Krysko, V.A.
Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems. *Entropy* **2018**, *20*, 175.
https://doi.org/10.3390/e20030175

**AMA Style**

Awrejcewicz J, Krysko AV, Erofeev NP, Dobriyan V, Barulina MA, Krysko VA.
Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems. *Entropy*. 2018; 20(3):175.
https://doi.org/10.3390/e20030175

**Chicago/Turabian Style**

Awrejcewicz, Jan, Anton V. Krysko, Nikolay P. Erofeev, Vitalyj Dobriyan, Marina A. Barulina, and Vadim A. Krysko.
2018. "Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems" *Entropy* 20, no. 3: 175.
https://doi.org/10.3390/e20030175