# 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

- Al-Naimee, K.; Marino, F.; Ciszak, M.; Meucci, R.; Arecchi, F.T. Chaotic spiking and incomplete homoclinic scenarios in semiconductor lasers with optoelectronic feedback. New J. Phys.
**2009**, 11, 073022. [Google Scholar] [CrossRef][Green Version] - Poincaré, H. Les Méthodes Nouvelles de la Mécanique Céleste; Gauthier-Villars: Paris, France, 1892; Volumes I–III. [Google Scholar]
- Andronov, A.A.; Chaikin, S.E. Theory of Oscillators, Moscow, I., English Translation; Princeton University Press: Princeton, NJ, USA, 1949. [Google Scholar]
- Levinson, N. A second-order differential equation with singular solutions. Ann. Math.
**1949**, 50, 127–153. [Google Scholar] [CrossRef] - Tikhonov, A.N. On the dependence of solutions of differential equations on a small parameter. Mat. Sb. N.S.
**1948**, 31, 575–586. [Google Scholar] - Wasow, W.R. Asymptotic Expansions for Ordinary Differential Equations; Wiley-Interscience: New York, NY, USA, 1965. [Google Scholar]
- Cole, J.D. Perturbation Methods in Applied Mathematics; Blaisdell: Waltham, MA, USA, 1968. [Google Scholar]
- O’Malley, R.E. Introduction to Singular Perturbations; Academic Press: New York, NY, USA, 1974. [Google Scholar]
- O’Malley, R.E. Singular Perturbations Methods for Ordinary Differential Equations; Springer: New York, NY, USA, 1991. [Google Scholar]
- Fenichel, N. Persistence and Smoothness of Invariant Manifolds for Flows. Ind. Univ. Math. J.
**1971**, 21, 193–225. [Google Scholar] [CrossRef] - Fenichel, N. Asymptotic stability with rate conditions. Ind. Univ. Math. J.
**1974**, 23, 1109–1137. [Google Scholar] [CrossRef] - Fenichel, N. Asymptotic stability with rate conditions II. Ind. Univ. Math. J.
**1977**, 26, 81–93. [Google Scholar] [CrossRef] - Fenichel, N. Geometric singular perturbation theory for ordinary differential equations. J. Diff. Eq.
**1979**, 31, 53–98. [Google Scholar] [CrossRef][Green Version] - Hirsch, M.W.; Pugh, C.C.; Shub, M. Invariant Manifolds; Springer: New York, NY, USA, 1977. [Google Scholar]
- Rossetto, B. Trajectoires lentes des syst‘emes dynamiques lents-rapides. In Analysis and Optimization of System; Springer: Berlin/Heidelberg, Germany, 1986; pp. 680–695. [Google Scholar]
- Rossetto, B. Singular approximation of chaotic slow-fast dynamical systems. In The Physics of Phase Space Nonlinear Dynamics and Chaos Geometric Quantization, and Wigner Function; Springer: Berlin/Heidelberg, Germany, 1987; pp. 12–14. [Google Scholar]
- Gear, C.W.; Kaper, T.J.; Kevrekidis, I.G.; Zagaris, A. Projecting to a slow manifold: Singularly perturbed systems and legacy codes. SIAM J. Appl. Dyn. Syst. Math.
**2005**, 4, 711–732. [Google Scholar] [CrossRef][Green Version] - Zagaris, A.; Gear, C.W.; Kaper, T.J.; Kevrekidis, Y.G. Analysis of the accuracy and convergence of equation-free projection to a slow manifold. ESAIM Math. Model. Num.
**2009**, 43, 757–784. [Google Scholar] [CrossRef] - Maas, U.; Pope, S.B. Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space. Combust. Flame
**1992**, 8, 239–264. [Google Scholar] [CrossRef] - Brøns, M.; Bar-Eli, K. Asymptotic analysis of canards in the EOE equations and the role of the inflection line. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**1994**, 445, 305–322. [Google Scholar] - Rossetto, B.; Lenzini, T.; Ramdani, S.; Suchey, G. Slow fast autonomous dynamical systems. Int. J. Bifurc. Chaos
**1998**, 8, 2135–2145. [Google Scholar] [CrossRef] - Ginoux, J.M.; Rossetto, B. Slow manifold of a neuronal bursting model. In Emergent Properties in Natural and Articial Dynamical Systems; Aziz-Alaoui, M.A., Bertelle, C., Eds.; Springer: Berlin/Heidelberg, Germany, 2006; pp. 119–128. [Google Scholar]
- Ginoux, J.M.; Rossetto, B. Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems. Int. J. Bif. Chaos
**2006**, 4, 887–910. [Google Scholar] [CrossRef][Green Version] - Ginoux, J.M.; Rossetto, B.; Chua, L.O. Slow Invariant Manifolds as Curvature of the Flow of Dynamical Systems. Int. J. Bif. Chaos
**2008**, 11, 3409–3430. [Google Scholar] [CrossRef][Green Version] - Ginoux, J.M. Differential Geometry Applied to Dynamical Systems; World Scientific Series on Nonlinear Science; Series A 66; World Scientific: Singapore, 2009. [Google Scholar]
- Ginoux, J.M.; Llibre, J. The flow curvature method applied to canard explosion. J. Phys. A Math. Theor.
**2011**, 44, 465203. [Google Scholar] [CrossRef] - Ginoux, J.M. The Slow Invariant Manifold of the Lorenz-Krishnamurthy Model. Qual. Theory Dyn. Syst.
**2014**, 13, 19–37. [Google Scholar] [CrossRef][Green Version] - Ginoux, J.M. Slow Invariant Manifolds of Slow-Fast Dynamical Systems. Int. J. Bif. Chaos
**2021**, 31, 2150112-1-17. [Google Scholar] [CrossRef] - Bender, C.M.; Orszag, S.A. Advanced Mathematical Methods for Scientists and Engineers; Springer: New York, NY, USA, 1999. [Google Scholar]
- Sandri, M. Numerical Calculation of Lyapunov Exponents. Math. J.
**1996**, 6, 78–84. [Google Scholar] - Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov Exponents from a Time Series. Phys. D
**1985**, 16, 285–317. [Google Scholar] [CrossRef][Green Version] - Eckmann, J.P.; Ruelle, D. Ergodic Theory of Chaos and Strange Attractors. Rev. Mod. Phys.
**1985**, 57, 617–656. [Google Scholar] [CrossRef] - Klein, M.; Baier, G. Hierarchies of Dynamical Systems. In A Chaotic Hierarchy; Baier, G., Klein, M., Eds.; World Scientific: Singapore, 1991. [Google Scholar]
- Ginoux, J.M.; Llibre, J.; Chua, L.O. Canards from Chua’s circuit. Int. J. Bif. Chaos
**2013**, 23, 1330010. [Google Scholar] [CrossRef][Green Version] - Ginoux, J.M.; Llibre, J. Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models. Math. Probl. Eng.
**2015**, 2015, 342010. [Google Scholar] [CrossRef][Green Version] - Ginoux, J.M.; Llibre, J. Canards Existence in Memristor’s Circuits. Qual. Theory Dyn. Syst.
**2016**, 15, 383–431. [Google Scholar] [CrossRef][Green Version] - Ginoux, J.M.; Llibre, J.; Tchizawa, K. Canards Existence in The Hindmarsh-Rose Model. Math. Model. Nat. Phenom.
**2019**, 14, 1–21. [Google Scholar] [CrossRef] - Darboux, G. Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré. Bull. Sci. Math. Sér.
**1878**, 2, 60–96, 123–143, 151–200. [Google Scholar]

**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