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  • 1
    Electronic Resource
    Electronic Resource
    Nonlinear dynamics Synchronization from chaos (2003)
    [s.l.] : Nature Publishing Group
    Nature 422 (2003), S. 384-385 
    Details
    ISSN: 1476-4687
    Source: Nature Archives 1869 - 2009
    Topics: Biology , Chemistry and Pharmacology , Medicine , Natural Sciences in General , Physics
    Notes: [Auszug] Chaos and control are often seen as opposite poles of the spectrum. But the theory of how to control dynamical chaos is evolving, and, in Physical Review Letters, Wei, Zhan and Lai present a welcome contribution. Chaos is a feature in all sciences: from lasers and meteorological systems, to ...
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1038/422384a
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  • 2
    Electronic Resource
    Electronic Resource
    Cycles homoclinic to chaotic sets; robustness and resonance (1997)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 7 (1997), S. 207-220 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: For dynamical systems possessing invariant subspaces one can have a robust homoclinic cycle to a chaotic set. If such a cycle is stable, it manifests itself as long periods of quiescent chaotic behaviour interrupted by sudden transient ‘bursts'. The time between the transients increases as the trajectory approaches the cycle. This behavior for a cycle connecting symmetrically related chaotic sets has been called ‘cycling chaos' by Dellnitz et al. [IEEE Trans. Circ. Sys. I 42, 821–823 (1995)]. We characterise such cycles and their stability by means of normal Lyapunov exponents. We find persistence of states that are not Lyapunov stable but still attracting, and also states that are approximately periodic. For systems possessing a skew-product structure (such as naturally arises in chaotically forced systems) we show that the asymptotic stability and the attractivity of the cycle depends in a crucial way on what we call the footprint of the cycle. This is the spectrum of Lyapunov exponents of the chaotic invariant set in the expanding and contracting directions of the cycle. Numerical simulations and calculations for an example system of a homoclinic cycle parametrically forced by a Rössler attractor are presented; here we observe the creation of nearby chaotic attractors at resonance of transverse Lyapunov exponents. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.166221
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  • 3
    Electronic Resource
    Electronic Resource
    Nonlinear dynamics When instability makes sense (2005)
    [s.l.] : Nature Publishing Group
    Nature 436 (2005), S. 36-37 
    Details
    ISSN: 1476-4687
    Source: Nature Archives 1869 - 2009
    Topics: Biology , Chemistry and Pharmacology , Medicine , Natural Sciences in General , Physics
    Notes: [Auszug] Dynamical stability is ubiquitous in many systems — and more often than not is desirable. Travelling down a straight road, a cyclist with stable dynamics will continue in more or less a straight line despite a gust of wind or a bumpy surface. In recent years, however, unstable dynamics has ...
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1038/436036b
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  • 4
    Electronic Resource
    Electronic Resource
    Symmetry groups of attractors (1994)
    Springer
    Archive for rational mechanics and analysis 126 (1994), S. 59-78 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract For maps equivariant under the action of a finite group Γ on ℝn, the possible symmetries of fixed points are known and correspond to the isotropy subgroups. This paper investigates the possible symmetries of arbitrary, possibly chaotic, attractors and finds that the necessary conditions of Melbourne, Dellnitz & Golubitsky [15] are sufficient, at least for continuous maps. This result shows that the reflection hyperplanes are important in determining those groups which are admissible; more precisely, a subgroup Σ of Γ is admissible as the symmetry group of an attractor if there exists a Δ with Σ/Δ cyclic such that Δ fixes a connected component of the complement of the set of reflection hyperplanes of reflections in Γ but not in Δ. For finite reflection groups this condition on Δ reduces to the condition that Δ is an isotropy subgroup. Our results are illustrated for finite subgroups of O(3).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00375696
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  • 5
    Electronic Resource
    Electronic Resource
    Heteroclinic Networks in Coupled Cell Systems (1999)
    Springer
    Archive for rational mechanics and analysis 148 (1999), S. 107-143 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract . We give an intrinsic definition of a heteroclinic network as a flow-invariant set that is indecomposable but not recurrent. Our definition covers many previously discussed examples of heteroclinic behavior. In addition, it provides a natural framework for discussing cycles between invariant sets more complicated than equilibria or limit cycles. We allow for cycles that connect chaotic sets (cycling chaos) or heteroclinic cycles (cycling cycles). Both phenomena can occur robustly in systems with symmetry. We analyze the structure of a heteroclinic network as well as dynamics on and near the network. In particular, we introduce a notion of ‘depth’ for a heteroclinic network (simple cycles between equilibria have depth 1), characterize the connections and discuss issues of attraction, robustness and asymptotic behavior near a network. We consider in detail a system of nine coupled cells where one can find a variety of complicated, yet robust, dynamics in simple polynomial vector fields that possess symmetries. For this model system, we find and prove the existence of depth‐2 networks involving connections between heteroclinic cycles and equilibria, and study bifurcations of such structures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002050050158
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  • 6
    Electronic Resource
    Electronic Resource
    A Hopf bifurcation with Robin boundary conditions (1994)
    Springer
    Journal of dynamics and differential equations 6 (1994), S. 487-505 
    Details
    ISSN: 1572-9222
    Keywords: Liapunov-Schmidt method ; Robin boundary condition ; Hopf bifurcation with square symmetry ; spatiotemporal pattern formation ; 35B32 ; 58G28 ; 65J15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate Hopf bifurcation of an example reaction-diffusion system on a square domain with Robin boundary conditions; the Brusselator equations. By performing a smooth homotopy of boundary conditions from Neumann to Dirichlet type, we observe the creation of branches of periodic solutions with submaximal symmetry in codimension two bifurcations, although we do not fully calculate the branching behaviour. We also note that mode interactions behave generically on varying the boundary conditions. The investigation is performed using a numerical Liapunov-Schmidt reduction technique of Ashwin, Böhmer, and Mei (1994) and an analysis of Swift (1988).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02218859
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  • 7
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    Lessons on Climate Sensitivity From Past Climate Changes (2016)
    Springer
    In:  EPIC3Current Climate Change Reports, Springer, 2(4), pp. 148-158, ISSN: 2198-6061
    Details
    Publication Date: 2019-10-01
    Description: Over the last decade, our understanding of cli- mate sensitivity has improved considerably. The climate system shows variability on many timescales, is subject to non-stationary forcing and it is most likely out of equi- librium with the changes in the radiative forcing. Slow and fast feedbacks complicate the interpretation of geolog- ical records as feedback strengths vary over time. In the geological past, the forcing timescales were different than at present, suggesting that the response may have behaved differently. Do these insights constrain the climate sensitiv- ity relevant for the present day? In this paper, we review the progress made in theoretical understanding of climate sensitivity and on the estimation of climate sensitivity from proxy records. Particular focus lies on the background state dependence of feedback processes and on the impact of tipping points on the climate system. We suggest how to further use palaeo data to advance our understanding of the currently ongoing climate change.
    Repository Name: EPIC Alfred Wegener Institut
    Type: Article , isiRev
    Format: application/pdf
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