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Orbit Classification and Sensitivity Analysis in Dynamical Systems Using Surrogate Models^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Hamiltonian Systems

#### 2.2. Symplectic Gaussian Process Emulation

#### 2.3. Sensitivity Analysis

#### 2.4. Local Lyapunov Exponents

## 3. Results and Discussion

#### 3.1. Local Lyapunov Exponents and Orbit Classification

#### 3.2. Sensitivity Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Distribution of local Lyapunov exponents for a (

**a**) regular orbit $(q,p)=(1.96,4.91)$ and (

**b**) chaotic orbit $(q,p)=(0.39,2.85)$ in the standard map with $K=2.0$.

**Figure 2.**Rate of convergence of the block bias due to finite number of mapping iterations for (

**a**) $K=2.0$ with a regular orbit $(q,p)=(1.96,4.91)$ (diamond) and a chaotic orbit $(q,p)=(0.39,2.85)$ (x) and (

**b**) $K=0.9$ with a regular orbit $(q,p)=(1.76,0.33)$ (diamond), a confined chaotic orbit $(q,p)=(0.02,2.54)$ (circle) and a chaotic orbit $(q,p)=(0.2,5.6)$ (x). The graphs show ${\tilde{\lambda}}_{T}$, the median of ${\lambda}_{T}$ for each T, with ${\tilde{\lambda}}_{T}=\lambda +c/T$ fitted by linear regression of $T{\tilde{\lambda}}_{T}$ on T. The gray areas correspond to the standard deviation for 1000 test points.

**Figure 3.**Local Lyapunov exponents in phase space of the standard map calculated with $T=50$ mapping iterations for (

**a**) $K=2.0$, (

**b**) $K=0.9$.

**Figure 4.**Orbit classification in standard map, (

**a**) $K=2.0$, (

**b**) $K=0.9$ for $T=50$. The color map indicates the probability that the orbit is regular.

**Figure 5.**Percentage of misclassified orbits using a Bayesian classifier trained with 200 orbits for (

**a**) $K=2.0$ and (

**b**) $K=0.9$. 100 test orbits on an equally spaced grid in the range of $[0,\pi ]\times [0,2\pi ]$ are classified as regular or chaotic depending on their LLE.

**Figure 6.**Total Sobol’ indices as a function of time for three orbits of the standard map with $K=0.9$—upper: chaotic orbit $(q,p)=(0.2,5.6)$, middle: regular orbit $(q,p)=(1.76,0.33)$, lower: regular orbit very close to fixed point $(q,p)=(\pi ,0.1)$.

**Figure 7.**Total Sobol’ indices (Equation (15)) for the standard map with $K=0.9$ averaged from $t=20$ to $t=30$.

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**MDPI and ACS Style**

Rath, K.; Albert, C.G.; Bischl, B.; von Toussaint, U.
Orbit Classification and Sensitivity Analysis in Dynamical Systems Using Surrogate Models. *Phys. Sci. Forum* **2021**, *3*, 5.
https://doi.org/10.3390/psf2021003005

**AMA Style**

Rath K, Albert CG, Bischl B, von Toussaint U.
Orbit Classification and Sensitivity Analysis in Dynamical Systems Using Surrogate Models. *Physical Sciences Forum*. 2021; 3(1):5.
https://doi.org/10.3390/psf2021003005

**Chicago/Turabian Style**

Rath, Katharina, Christopher G. Albert, Bernd Bischl, and Udo von Toussaint.
2021. "Orbit Classification and Sensitivity Analysis in Dynamical Systems Using Surrogate Models" *Physical Sciences Forum* 3, no. 1: 5.
https://doi.org/10.3390/psf2021003005