Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing

Broer, Hendrik; Simó, Carles; Vitolo, Renato

doi:10.1088/0951-7715/15/4/312

Cited by 104 publications

(102 citation statements)

References 62 publications

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“…We conclude by observing that the complexity of the present bifurcation can be easily met in concrete studies, e.g., see [15] where this and related problems are encountered in the dynamical modelling of the northern hemisphere climate.…”

Section: Theoretical Expectationsmentioning

confidence: 68%

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Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Broer

Simó

Vitolo

2008

Physica D: Nonlinear Phenomena

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The dynamics near a Hopf-saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf-saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.

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Section: Theoretical Expectationsmentioning

confidence: 68%

“…A HSN bifurcation of fixed points is one of the organising centres of the bifurcation diagram of a diffeomorphism arising in the study of a climatological model, see [15] and [60,Chap 2].…”

Section: Introductionmentioning

confidence: 99%

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Broer

Simó

Vitolo

2008

Physica D: Nonlinear Phenomena

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“…Despite the simplicity of the Lorenz-84 model, it addresses many key applications in climate studies such as how the coexistence of two possible climates combined with variations of the solar heating causes seasons with inter-annual variability [6,58,59,67], how the climate is affected by the interactions atmosphere and the oceans [7,70], how the asymmetry between oceans and continents may result in complex behaviors of the system [62], etc. In addition to applications, the Lorenz-84 model has also attracted much attentions from mathematicians because of certain interesting and subtle mathematical aspects of its underlying differential equations such as multistability, intransitivity and bifurcation [35].…”

Section: Application To Climate Change: the Lorenz-84 Modelmentioning

confidence: 99%

Applied Nonautonomous and Random Dynamical Systems

Caraballo¹,

Han²

2016

SpringerBriefs in Mathematics

114

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PrefaceDuring the past two decades, the theory of nonautonomous dynamical systems and random dynamical systems has made substantial progress in studying the long term dynamics of open systems subject to time-dependent or random forcing. However, most of the existing pertinent literatures are fairly technical and almost impenetrable for a general audience, except the most dedicated specialists. Moreover, the concepts and methods of nonautonomous and random dynamical systems, though well-established, are extremely nontrivial to apply to real-world problems. The aim of this work is to provide an accessible and broad introduction to the theory of nonautonomous and random dynamical systems, with an emphasis on applications of the theory to problems arising in the applied sciences and engineering.The book starts with basic concepts in the theory of autonomous dynamical systems which are easier to understand, and used as the motivation for the study of non-autonomous and random dynamical systems. Then the framework of nonautonomous dynamical systems is set up, including various approaches to analyze the long time behavior of non-autonomous problems. The major emphasis is given to the novel theory of pullback attractors, as it can be regarded as a natural extension of the autonomous theory and allows a larger variety of time-dependence forcing than other alternatives such as skew-product flows or cocycles. In the end the theory of random dynamical systems and random attractors is introduced and shown to fairly informative to the study of long term behavior of stochastic systems with random forcing.Each set of theory is illustrated by being applied to three different models, the chemostat model, the SIR epidemic model, and the Lorenz-84 model, in their autonomous, nonautonomous, and stochastic formulations, respectively. The techniques and methods adopted can be applied to the study of the long term behavior of a wide range of applications arising in applied sciences and engineering.

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“…Analytical studies of bifurcation phenomena and their normal forms are sometimes supported by numerical computations, e.g., in [37,8]. In such studies, curves of codim 1 bifurcations of limit cycles are computed (often with AUTO [17]), and codim 2 bifurcations are detected.…”

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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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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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Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing

Cited by 104 publications

References 62 publications

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

Applied Nonautonomous and Random Dynamical Systems

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

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