Introduction to bifurcation theory

Crawford, John D.

doi:10.1103/revmodphys.63.991

Cited by 327 publications

(217 citation statements)

References 62 publications

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“…Equation (23) describes a pitchfork bifurcation (Crawford, 1991) from the constant density state to a spatially-periodic pattern with steady-state amplitude determined by |z| 2 = −λm/a. For K = K e 1 from (10a), a > 0 and so the patterned state bifurcates subcritically, and hence unstably.…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

A Nonlocal Continuum Model for Biological Aggregation

Bertozzi²,

2006

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We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. For the case of one spatial dimension, we study the steady states analytically and numerically. There exist strongly nonlinear states with compact support and steep edges that correspond to localized biological aggregations, or clumps. These steady state clumps are approached through a dynamic coarsening process. In the limit of large population size, the clumps approach a constant density swarm with abrupt edges. We use energy arguments to understand the nonlinear selection of clump solutions, and to predict the internal density in the large population limit. The energy result holds in higher dimensions, as well, and is demonstrated via numerical simulations in two dimensions.

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Section: Weakly Nonlinear Analysismentioning

confidence: 99%

A Nonlocal Continuum Model for Biological Aggregation

Bertozzi²,

2006

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“…The "parameters" will usually evolve on time scales slower than the "state variables". This separation of time-scales is an important concept in physics and engineering to decide which variables are considered static and which dynamic (Crawford, 1991;Haken, 1983).…”

Section: Reactive: Temporary Entrainment and Shape Changesmentioning

confidence: 99%

Engineering entrainment and adaptation in limit cycle systems

2006

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Periodic behavior is key to life, and is observed in multiple instances and at multiple time scales in our metabolism, our natural environment, and our engineered environment. A natural way of modelling or generating periodic behavior is done by using oscillators, i.e. dynamical systems that exhibit limit cycle behavior. While there is extensive literature on methods to analyze such dynamical systems, much less work has been done on methods to synthesize an oscillator to exhibit some specific desired characteristics. The goal of this article is two-fold: (1) to provide a framework for characterizing and designing oscillators, and (2) to review how classes of well known oscillators can be understood and related to this framework.The basis of the framework is to characterize oscillators in terms of their fundamental temporal and spatial behavior, and in terms of properties that these two behaviors can be designed to exhibit. This focus on fundamental properties is important because it allows us to systematically compare a large variety of oscillators which might at first sight appear very different from each other. We identify several specifications that are useful for design, such as frequency-locking behavior, phaselocking behavior, and specific output signal shape. We also identify two classes of design methods by which these specifications can be met, namely off-line methods and on-line methods. By relating these specifications to our framework and by presenting several examples of how oscillators have been designed in the literature, this article provides a useful methodology and toolbox for designing oscillators for a wide range of purposes. In particular the focus on synthesis of limit cycle dynamical systems should be useful both for engineering and for computational modelling of physical or biological phenomena.

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“…This is the structural stability of systems as studied in dynamical system theory (Guckenheimer and Holmes 1983;Crawford 1991). We will show that there exists a parameter range in which globally attracting equilibria of the wave-mean flow system exist, and that as a parameter changes these become unstable through a bifurcation leading to periodic vacillation of the system and eventually, as the parameter further increases, to chaos.…”

Section: J O U R N a L O F T H E A T M O S P H E R I C S C I E N C E Smentioning

confidence: 99%

Structural Stability of Turbulent Jets

Farrell¹,

Ioannou²

2003

J. Atmos. Sci.

132

204

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Turbulence in fluids is commonly observed to coexist with relatively large spatial and temporal scale coherent jets. These jets may be steady, vacillate with a definite period, or be irregular. A comprehensive theory for this phenomenon is presented based on the mutual interaction between the coherent jet and the turbulent eddies. When a sufficient number of statistically independent realizations of the eddy field participate in organizing the jet a simplified asymptotic dynamics emerges with progression, as an order parameter such as the eddy forcing is increased, from a stable fixed point associated with a steady symmetric zonal jet through a pitchfork bifurcation to a stable asymmetric jet followed by a Hopf bifurcation to a stable limit cycle associated with a regularly vacillating jet and finally a transition to chaos. This underlying asymptotic dynamics emerges when a sufficient number of ensemble members is retained in the stochastic forcing of the jet but a qualitative different mean jet dynamics is found when a small number of ensemble members is retained as is appropriate for many physical systems. Example applications of this theory are presented including a model of midlatitude jet vacillation, emergence and maintenance of multiple jets in turbulent flow, a model of rapid reorganization of storm tracks as a threshold in radiative forcing is passed, and a model of the quasi-biennial oscillation. Because the statistically coupled wave-mean flow system discussed is generally globally stable this system also forms the basis for a comprehensive theory for equilibration of unstable jets in turbulent shear flow.

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Introduction to bifurcation theory

Cited by 327 publications

References 62 publications

A Nonlocal Continuum Model for Biological Aggregation

A Nonlocal Continuum Model for Biological Aggregation

Engineering entrainment and adaptation in limit cycle systems

Structural Stability of Turbulent Jets

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