# Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Lyapunov Characteristic Exponents

#### 3.1. Theoretical Overview

#### 3.2. Numerical Approach

## 4. N-Body Simulation

#### 4.1. Equations of Motion

#### 4.2. Time Evolution of Tangent-Space Vectors

#### 4.2.1. Coulombic System

#### 4.2.2. Gravitational System

#### 4.3. p-Volume and Lyapunov Spectrum

#### 4.4. Results

## 5. Discussion

- (
**a**) - The sum of LCEs was found to converge to zero for all energies and N.
- (
**b**) - For the ordered set $\left\{{\lambda}_{i}\right\}$ (${\lambda}_{1}\ge {\lambda}_{2}\ge \cdots \ge {\lambda}_{2N}$), the results show that$${\lambda}_{i}\sim -{\lambda}_{2N-i+1},\phantom{\rule{15.0pt}{0ex}}i=1,2,\cdots ,2N.$$
- (
**c**) - In addition, our results show that$${\lambda}_{N-1}\sim {\lambda}_{N}\sim {\lambda}_{N+1}\sim {\lambda}_{N+2}\sim 0.$$

**c**) is nothing but a consequence of the conservation of momentum [25].

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

LCE | Lyapunov characterstic exponent |

KS | Kolmogorov–Sinai |

MD | Molecular dynamics |

HMF | Hamiltonian Mean Field |

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**Figure 1.**Full spectra of Lyapunov characteristic exponents (LCEs) plotted against per-particle energy for (

**a**) Coulombic system, and (

**b**) gravitational system, with $N=11$. The topmost curve shows ${\lambda}_{1}$, the second to top curve shows ${\lambda}_{2}$, and so on all the way to the curve on the very bottom representing ${\lambda}_{22}$ in (

**a**,

**b**). The central solid (blue) line in each plot shows the sum of LCEs. ${\mathcal{H}}_{c}$ and ${\mathcal{H}}_{g}$ are expressed in units of $\frac{2L}{N}\left|\kappa \right|$. ${\lambda}_{i}$ are expressed in units of (

**a**) $\omega $, and (

**b**) $\mathsf{\Lambda}$.

**Figure 2.**Energy dependence of (

**a**) the largest LCE, and (

**b**) Kolmogorov-entropy density for Coulombic system with different degrees of freedom. ${\lambda}_{1}$ and ${\lambda}_{S}$ are expressed in units of $\omega $, whereas ${\mathcal{H}}_{c}$ is expressed in units of $\frac{2L}{N}\left|\kappa \right|$.

**Figure 3.**Energy dependence of (

**a**) the largest LCE, and (

**b**) Kolmogorov-entropy density for gravitational system with different degrees of freedom. ${\lambda}_{1}$ and ${\lambda}_{S}$ are expressed in units of $\mathsf{\Lambda}$, whereas ${\mathcal{H}}_{g}$ is expressed in units of $\frac{2L}{N}\left|\kappa \right|$.

**Figure 4.**Energy dependence of the normalized values of the largest LCE and Kolmogorov-entropy density for (

**a**) Coulombic system, and (

**b**) gravitational system for $N=11$. ${\mathcal{H}}_{c}$ and ${\mathcal{H}}_{g}$ are expressed in units of $\frac{2L}{N}\left|\kappa \right|$, whereas $\widehat{\lambda}$ is dimensionless.

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Kumar, P.; Miller, B.N.
Lyapunov Spectra of Coulombic and Gravitational Periodic Systems. *Entropy* **2017**, *19*, 238.
https://doi.org/10.3390/e19050238

**AMA Style**

Kumar P, Miller BN.
Lyapunov Spectra of Coulombic and Gravitational Periodic Systems. *Entropy*. 2017; 19(5):238.
https://doi.org/10.3390/e19050238

**Chicago/Turabian Style**

Kumar, Pankaj, and Bruce N. Miller.
2017. "Lyapunov Spectra of Coulombic and Gravitational Periodic Systems" *Entropy* 19, no. 5: 238.
https://doi.org/10.3390/e19050238