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The Slow Invariant Manifold of the Lorenz–Krishnamurthy Model

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Abstract

During this last decades, several attempts to construct slow invariant manifold of the Lorenz–Krishnamurthy five-mode model of slow–fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic procedures involved for such a construction diverge. So, it seems that till now only the first-order and third-order approximations of this slow manifold have been analytically obtained. While using the Flow Curvature Method we show in this work that one can provide the eighteenth-order approximation of the slow manifold of the generalized Lorenz–Krishnamurthy model and the thirteenth-order approximation of the “conservative” Lorenz–Krishnamurthy model. The invariance of each slow manifold is then established according to Darboux invariance theorem.

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Notes

  1. This theory was independently established in Hirsch et al. [19].

  2. This concept has been introduced by Leith [24] in 1980.

  3. In certain applications these functions will be supposed to be \(C^r\), \(r \geqslant 1\).

  4. Boyd [2, p. 1058].

  5. See http://ginoux.univ-tln.fr for complete equation.

  6. See http://ginoux.univ-tln.fr for complete equation.

  7. By considering that each vector \(\vec {a}_i \) may be spanned on the eigenbasis, calculus is longer and tedious but leads to the same result.

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Acknowledgments

Authors would like to thank Prof. Jaume Llibre for his mathematical remarks and comments.

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Authors and Affiliations

  1. Laboratoire LSIS, CNRS, UMR 7297, Université de Toulon, BP 20132, 83957 , La Garde Cedex, France

    Jean-Marc Ginoux

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Appendix

Appendix

The identity involved in the proof of the invariance of the slow manifold (Sect. 4.2.2) is stated in this appendix.

$$\begin{aligned}&J\vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3\wedge \ldots \wedge \vec {a}_n } \right) +\vec {a}_1 \cdot \left( {J\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \vec {a}_n } \right) \nonumber \\&\quad +\ldots +\vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge J\vec {a}_n } \right) = Tr\left( J \right) \vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \ldots \wedge \vec {a}_n } \right) \end{aligned}$$
(42)

Proof

The proof is based on inner product properties.

To the functional jacobian matrix \(J\) is associated an eigenbasis: \(\{ \vec {Y_{\lambda _1 }} ,\vec {Y_{\lambda _2 }}, \ldots ,\vec {Y_{\lambda _n}}\}\).

Let suppose that there exists a transformationFootnote 7 such that: to each vector \(\vec {a}_i \) corresponds the eigenvector \(\vec {Y_{\lambda _i }}\) with \(i=1,\ldots ,n\).

Each inner product of the left hand side Eq. (42) may be transformed into

$$\begin{aligned}&J\vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \vec {a}_n } \right) =\lambda _1 \vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \vec {a}_n } \right) =\lambda _1 \vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \vec {a}_n } \right) \\&\vec {a}_1 \cdot \left( {J\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \vec {a}_n } \right) = \vec {a}_1 \cdot \left( {\lambda _2 \vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \vec {a}_n } \right) = \lambda _2 \vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \vec {a}_n } \right) \\&{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots }{\ldots } \\&\vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge J\vec {a}_n } \right) = \vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \lambda _n \vec {a}_n } \right) = \lambda _n \vec {a}_1 \cdot \left( {\vec {a}_2 \wedge \vec {a}_3 \wedge \ldots \wedge \vec {a}_n } \right) \end{aligned}$$

Making the sum of these factors the proof is stated.

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Ginoux, JM. The Slow Invariant Manifold of the Lorenz–Krishnamurthy Model. Qual. Theory Dyn. Syst. 13, 19–37 (2014). https://doi.org/10.1007/s12346-013-0104-6

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