# Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems

## Abstract

**:**

## 1. Introduction

## 2. Motivation

## 3. Overview of Khinchin’s Entropy

- Calculating the equi-energy volume enclosed by the system’s closed-surface trajectory in phase space, V.
- Determining the structure function, $\Omega $.
- Determining the generating function, $\Phi $.
- Finding the value $s=\sigma $ that satisfies Equation (8).
- Substituting the generating function and $\sigma $ into Equation (7).

#### 3.1. Linear Systems

#### 3.2. Nonlinear Systems

#### 3.3. Mixing Entropy

## 4. Entropy of Duffing Oscillators

#### 4.1. A Duffing Oscillator

#### 4.1.1. The Phase Volume

#### 4.1.2. Khinchin’s Entropy

#### 4.2. Coupled Duffing Oscillators

#### 4.2.1. Entropy of the Coupled Duffing System

#### 4.2.2. Mixing Entropy of the Coupled Duffing System

## 5. Entropy of Henon–Heiles Oscillators

#### 5.1. Single Degree of Freedom Oscillator with Third Order Anharmonic Potential

#### 5.1.1. The Phase Volume

#### 5.1.2. Khinchin’s Entropy

#### 5.2. Henon–Heiles Oscillators

#### 5.2.1. Entropy of the Henon–Heiles Oscillators

#### 5.2.2. Mixing Entropy of the Henon–Heiles Oscillators

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${(.)}^{\prime}$ | Derivative of $(.)$ with respect to x |

$\dot{(.)}$ | Time-derivative of $(.)$ |

${x}_{i}$ | i-th displacement degree of freedom |

${x}_{M}$ | Maximum displacement over time |

${m}_{i}$ | i-th oscillator mass |

${k}_{i}$ | i-th spring stiffness |

${k}_{nl}$ | Nonlinear spring stiffness |

${k}_{c}$ | Coupling spring stiffness |

q | Generalized coordinate |

p | Generalized velocity |

$\nu $ | Nonlinearity multiplier |

${(.)}^{\left(i\right)}$ | i-th order approximation of $(.)$ with respect to $\nu $ |

${(.)}^{\left[i\right]}$ | i-th term in expansion of $(.)$ with respect to $\nu $ |

E | Energy |

T | Thermodynamic temperature |

$\Sigma $ | System trajectory in phase space |

V | Equi-energy volume in phase space |

$\Omega $ | Structure function |

$\Phi $ | Generating function |

H | Entropy |

${H}^{*}$ | Decoupled entropy |

${H}_{mix}$ | Mixing entropy |

$B,C$ | Scalar constants |

$\sigma $ | Value that minimizes H |

$:=$ | Equal by definition |

$\mathit{q}$ | Generalized coordinate matrix |

## Appendix A

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**Figure 2.**Closed-surface trajectory of a Duffing oscillator in phase space for several values of $\nu $ ranging from 0 to 6.

**Figure 3.**Comparison of the first through fourth order approximate volumes, ${V}^{\left(1\right)}$, ${V}^{\left(2\right)}$, ${V}^{\left(3\right)}$, and ${V}^{\left(4\right)}$, with the exact volume, V, for (

**a**) $\nu =0$ through $\nu =1$ and (

**b**) $\nu =0$ through $\nu =6$.

**Figure 4.**Comparison of percent error between the first through fourth order approximate volumes, ${V}^{\left(1\right)}$ through ${V}^{\left(4\right)}$, and the exact volume, V, for (

**a**) $\nu =0$ through $\nu =1$ and (

**b**) $\nu =0$ through $\nu =6$.

**Figure 5.**Comparison of (

**a**) the first through fourth order approximate volumes, ${V}^{\left(1\right)}$ through ${V}^{\left(4\right)}$, and the exact volume, V, and (

**b**) the error in the first through fourth order approximate volumes, for $\omega \approx 0.09$ through $\omega =3.0$.

**Figure 6.**Entropies found using the first and second order phase volume approximations, ${H}^{\left(1\right)}$ and ${H}^{\left(2\right)}$, compared to the linear entropy, ${H}_{L}$, for (

**a**) $\nu =0$ through $\nu =1$ and (

**b**) $\nu =0$ through $\nu =6$.

**Figure 8.**Energies ${E}_{1}$ and ${E}_{2}$ of the Duffing oscillators compared to their sum, ${E}_{1}+{E}_{2}$, and the total energy, E, for $\nu =0.2$.

**Figure 9.**Actual entropy, H, total hypothetical decoupled entropy, ${H}^{*}$, and mixing entropy, ${H}_{mix}$, of the coupled Duffing system approximated to the first and second order for $\nu =0.2$.

**Figure 10.**Closed-surface trajectory in phase space of an oscillator with velocity given by Equation (68). Trajectories are shown for several values of $\nu $ ranging from 0 to $0.1$. All values of $\nu $ are less than the critical value, ${\nu}_{c}$.

**Figure 11.**Position of the oscillator described by Equation (67), shown for $\nu $ just above, below, and equal to the critical value, ${\nu}_{c}$.

**Figure 12.**Phase portrait of the oscillator with quadratic repulsive force $\nu {k}_{nl}{x}^{2}$ described by Equation (67) for (

**a**) $\nu =0$ (

**b**) $\nu =0.25$ (

**c**) $\nu =0.4$ and (

**d**) $\nu ={\nu}_{c}=1/\sqrt{6}$. The dashed line represents the separatrix when $\nu ={\nu}_{c}$, the solid line is the system trajectory when $E=1$, and the arrows represent trajectories at other energies. Equilibrium points are shown as dots.

**Figure 13.**Comparison of ${V}^{\left(1\right)},{V}^{\left(2\right)},$ and ${V}^{\left(4\right)}$ for the oscillator described by Equation (67) with the exact volume, V, for (

**a**) $\nu =0$ through $\nu =0.1$ and (

**b**) $\nu =0$ through $\nu =0.4$.

**Figure 14.**Comparison of the percent errors from the exact volume, V, in ${V}^{\left(1\right)},\phantom{\rule{4pt}{0ex}}{V}^{\left(2\right)},$ and ${V}^{\left(4\right)}$ for the oscillator described by Equation (67) for (

**a**) $\nu =0$ through $\nu =0.2$ and (

**b**) $\nu =0$ through $\nu =0.4$.

**Figure 15.**Comparison of (

**a**) the first through fourth order approximate volumes, ${V}^{\left(1\right)}$ through ${V}^{\left(4\right)}$, and the exact volume, V, and (

**b**) the error in the first through fourth order approximate volumes for $\omega \approx 0.04$ through $\omega =1.0$

**Figure 16.**Entropies ${H}^{\left(2\right)}$ and ${H}^{\left(4\right)}$ found using second and fourth order phase volume approximations for (

**a**) $\nu =0$ through $\nu =0.1$ and (

**b**) $\nu =0$ through $\nu =0.4$.

**Figure 18.**Energies ${E}_{1}$ and ${E}_{2}$ of the Henon–Heiles oscillators compared to their sum, ${E}_{1}+{E}_{2}$, and the total energy, E.

**Figure 19.**Actual entropy, H, total hypothetical decoupled entropy, ${H}^{*}$, and mixing entropy, ${H}_{mix}$, of the Henon–Heiles system approximated to the second and fourth order.

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Sotoudeh, Z.
Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems. *Entropy* **2019**, *21*, 536.
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**AMA Style**

Sotoudeh Z.
Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems. *Entropy*. 2019; 21(5):536.
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**Chicago/Turabian Style**

Sotoudeh, Zahra.
2019. "Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems" *Entropy* 21, no. 5: 536.
https://doi.org/10.3390/e21050536