Frequency Scaling at the Onset of Finite Amplitude Oscillation

1986 ◽  
Vol 41 (12) ◽  
pp. 1412-1414 ◽  
Author(s):  
H. Bruce Stewart
Keyword(s):  
Scaling Law ◽  
Finite Amplitude ◽  
Bifurcation Parameter ◽  
Frequency Scaling ◽  
Amplitude Oscillation

The onset of oscillation at non-zero finite amplitude is shown to be governed in the generic case by a scaling law for the frequency of oscillation f in terms of the bifurcation parameter μ – μc namely f ~ (μ – μ)c1/2. This law can be expected to hold whenever such a bifurcation occurs in a nondegenerate system without hysteresis.

scholarly journals Finite Cascades of Pitchfork Bifurcations and Multistability in Generalized Lorenz-96 Models

2020 ◽  
Vol 25 (4) ◽  
pp. 78
Author(s):  
Anouk F. G. Pelzer ◽  
Alef E. Sterk
Keyword(s):  
Dynamical Systems ◽  
Scaling Law ◽  
Chaotic Attractors ◽  
Bifurcation Parameter ◽  
Coexisting Attractors ◽  
Pitchfork Bifurcations ◽  
Primary Focus ◽  
Divisibility Properties ◽  
Lorenz 96 ◽  
Parameter Values

In this paper, we study a family of dynamical systems with circulant symmetry, which are obtained from the Lorenz-96 model by modifying its nonlinear terms. For each member of this family, the dimension n can be arbitrarily chosen and a forcing parameter F acts as a bifurcation parameter. The primary focus in this paper is on the occurrence of finite cascades of pitchfork bifurcations, where the length of such a cascade depends on the divisibility properties of the dimension n. A particularly intriguing aspect of this phenomenon is that the parameter values F of the pitchfork bifurcations seem to satisfy the Feigenbaum scaling law. Further bifurcations can lead to the coexistence of periodic or chaotic attractors. We also describe scenarios in which the number of coexisting attractors can be reduced through collisions with an equilibrium.

scholarly journals Workload Frequency Scaling Law - Derivation and Verification

Queue ◽  
2018 ◽  
Vol 16 (2) ◽  
pp. 50-66
Author(s):  
Noor Mubeen
Keyword(s):  
Scaling Law ◽  
Frequency Scaling

scholarly journals Dynamo saturation down to vanishing viscosity: strong-field and inertial scaling regimes

2019 ◽  
Vol 864 ◽  
pp. 971-994 ◽  
Author(s):  
Kannabiran Seshasayanan ◽  
Basile Gallet
Keyword(s):  
Large Scale ◽  
Magnetic Energy ◽  
Strong Field ◽  
Scaling Law ◽  
Eddy Diffusivity ◽  
Finite Amplitude ◽  
Subcritical Case ◽  
Navier Stokes Equation ◽  
Low Viscosity ◽  
Magnetic Diffusivity

We present analytical examples of fluid dynamos that saturate through the action of the Coriolis and inertial terms of the Navier–Stokes equation. The flow is driven by a body force and is subject to global rotation and uniform sweeping velocity. The model can be studied down to arbitrarily low viscosity and naturally leads to the strong-field scaling regime for the magnetic energy produced above threshold: the magnetic energy is proportional to the global rotation rate and independent of the viscosity $\unicode[STIX]{x1D708}$. Depending on the relative orientations of global rotation and large-scale sweeping, the dynamo bifurcation is either supercritical or subcritical. In the supercritical case, the magnetic energy follows the scaling law for supercritical strong-field dynamos predicted on dimensional grounds by Pétrélis & Fauve (Eur. Phys. J. B, vol. 22, 2001, pp. 271–276). In the subcritical case, the system jumps to a finite-amplitude dynamo branch. The magnetic energy obeys a magneto-geostrophic scaling law (Roberts & Soward, Annu. Rev. Fluid Mech., vol. 4, 1972, pp. 117–154), with a turbulent Elsasser number of the order of unity, where the magnetic diffusivity of the standard Elsasser number appears to be replaced by an eddy diffusivity. In the absence of global rotation, the dynamo bifurcation is subcritical and the saturated magnetic energy obeys the equipartition scaling regime. We consider both the vicinity of the dynamo threshold and the limit of large distance from threshold to put these various scaling behaviours on firm analytical ground.

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