# Slow Invariant Manifold of Laser with Feedback

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## Abstract

**:**

## 1. Introduction

## 2. Slow–Fast Dynamical System

## 3. Stability Analysis

#### 3.1. Fixed Points, Jacobian Matrix and Eigenvalues

#### 3.2. Bifurcation Diagram

#### 3.3. Numerical Computation of the Lyapunov Characteristic Exponents

## 4. Slow Invariant Manifold

**Proposition**

**1.**

## 5. Discussion

“Therefore, it is expected that they should not imply strong modifications of the slow-manifold shape which, as discussed above, is responsible for the observed dynamics.”

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Slow invariant manifolds of system (1) in the phase space for various values of $\delta $.

$\mathit{\delta}$ | LCE Spectrum | Dynamics of the Attractor | Hausdorff Dimension |
---|---|---|---|

$\delta \in [1.0060,1.0082]$ | ($0,-,-$) | Limit cycle of period 1 | $D=1$ |

$\delta \in [1.0082,1.0105]$ | ($0,-,-$) | Limit cycle of period 2 | $D=1$ |

$\delta \in [1.0105,1.0117]$ | ($0,-,-$) | Limit cycle of period 4 | $D=1$ |

$\delta \in [1.0117,1.0137]$ | ($0,-,-$) | Limit cycle of period 2 | $D=1$ |

$\delta \in [1.0137,1.02]$ | ($+,0,-$) | 2-Chaos | $D=2.02$ |

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Ginoux, J.-M.; Meucci, R.
Slow Invariant Manifold of Laser with Feedback. *Symmetry* **2021**, *13*, 1898.
https://doi.org/10.3390/sym13101898

**AMA Style**

Ginoux J-M, Meucci R.
Slow Invariant Manifold of Laser with Feedback. *Symmetry*. 2021; 13(10):1898.
https://doi.org/10.3390/sym13101898

**Chicago/Turabian Style**

Ginoux, Jean-Marc, and Riccardo Meucci.
2021. "Slow Invariant Manifold of Laser with Feedback" *Symmetry* 13, no. 10: 1898.
https://doi.org/10.3390/sym13101898