Bifurcation and Predictability Analysis of a Low-Order Atmospheric Circulation Model

Shilnikov, Andrey; Nicolis, Grégoire; Nicolis, C.

doi:10.1142/s0218127495001253

Cited by 93 publications

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“…We fix a = 0.25 and b = 4. This model, as found in [40,43], has most of the analyzed codim 2 bifurcations of limit cycles. We report in Figure 6 bifurcation diagram recomputed and extended with MATCONT in which the bifurcations of equilibria (LP stands for limit point and H for Andronov-Hopf) are thicker and the limit cycle bifurcations are thin.…”

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confidence: 86%

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Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles: Computational Formulas, Numerical Implementation, and Examples

Witte¹,

Rossa²,

Govaerts³

et al. 2013

SIAM J. Appl. Dyn. Syst.

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Abstract. Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T ], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundaryvalue algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time.

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confidence: 86%

“…It can be checked that the equations for h 40 , h 50 , and h 41 are uniquely solvable. Since we are in a complex eigenvalue case, v is determined up to a factor γ, for whichγ…”

Section: Chenciner Bifurcationmentioning

confidence: 99%

Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles: Computational Formulas, Numerical Implementation, and Examples

Witte¹,

Rossa²,

Govaerts³

et al. 2013

SIAM J. Appl. Dyn. Syst.

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show abstract

“…This low-order model is not only appealing (Shil'nikov et al, 1995;Provenzale and Balmford, 1999;Tél and Gruiz, 2006, p. 293;Freire et al, 2008), but it can be derived from the quasi-geostrophic equations governing the large-scale motion of the atmosphere (Roebber, 1995). As a driving, we choose a paradigmatic chaotic signal, the first component of the classical Lorenz equations (L63) (Lorenz, 1963) coupled in an additive manner.…”

Section: Introductionmentioning

confidence: 99%

A chaotically driven model climate: extreme events and snapshot attractors

Bódai¹,

Károlyi

Tél³

2011

Nonlin. Processes Geophys.

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Abstract. In a low-order chaotic global atmospheric circulation model the effects of deterministic chaotic driving are investigated. As a result of driving, peak-over-threshold type extreme events, e.g. cyclonic activity in the model, become more extreme, with increased frequency of recurrence. When the characteristic time of the driving is comparable to that of the undriven system, a resonance effect with amplified variance shows up. For very fast driving we find a reduced enhancement of variance, which is also the case with white noise driving. Snapshot attractors and their natural measures are determined as a function of time, and a resonance effects is also identified. The extreme value statistics of group maxima is found to follow a Weibull distribution.

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“…So far, this model was known to have a pair of coexisting climates described originally by Lorenz [18]. Due to the importance of Lorenz-84 model, it has received great attention from researchers, and many important results on Lorenz-84 model have also been obtained [19][20][21][22][23]. In 1995, Shil'nikov et al discussed the bifurcation and predictability of the Lorenz-84 model [19].…”

Section: Introductionmentioning

confidence: 99%

Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model

Wen

2014

Journal of Applied Mathematics

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The dynamical behaviors of the Lorenz-84 atmospheric circulation model are investigated based on qualitative theory and numerical simulations. The stability and local bifurcation conditions of the Lorenz-84 atmospheric circulation model are obtained. It is also shown that when the bifurcation parameter exceeds a critical value, the Hopf bifurcation occurs in this model. Then, the conditions of the supercritical and subcritical bifurcation are derived through the normal form theory. Finally, the chaotic behavior of the model is also discussed, the bifurcation diagrams and Lyapunov exponents spectrum for the corresponding parameter are obtained, and the parameter interval ranges of limit cycle and chaotic attractor are calculated in further. Especially, a computerassisted proof of the chaoticity of the model is presented by a topological horseshoe theory.

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Bifurcation and Predictability Analysis of a Low-Order Atmospheric Circulation Model

Cited by 93 publications

References 0 publications

Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles: Computational Formulas, Numerical Implementation, and Examples

Numerical Periodic Normalization for Codim 2 Bifurcations of Limit Cycles: Computational Formulas, Numerical Implementation, and Examples

A chaotically driven model climate: extreme events and snapshot attractors

Dynamical Analysis of the Lorenz-84 Atmospheric Circulation Model

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