A manifest Hopf bifurcation in resistive magnetohydrodynamics
TL;DR: A special Hopf bifurcation in resistive magnetohydrodynamics can be identified by starting from a particularly suitable form of the linearized equations which was previously introduced by the author as mentioned in this paper.
About: This article is published in Physics Letters A.The article was published on 1993-09-06. It has received 3 citations till now. The article focuses on the topics: Saddle-node bifurcation & Hopf bifurcation.
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TL;DR: In this article, a sufficient condition for linear stability of dissipative mechanical systems with circulatory forces is derived, which has important applications in plasma physics, and it is shown that dissipative systems which are more than marginally stable without gyroscopic and circulatory force, remain stable after the addition of weak but finite gyroscopy and circulation forces.
10 citations
TL;DR: In this paper, a general stability condition for a class of real systems, occurring especially in plasma physics, is proved to persist to second order, despite the addition of several antisymmetric operators of first order in the linearized stability equation.
Abstract: Two general problems related to resistive magnetohydrodynamic stability are addressed in this paper. First, a general stability condition previously derived by the author for a class of real systems, occurring especially in plasma physics, is proved to persist to second order, despite the addition of several antisymmetric operators of first order in the linearized stability equation. Second, for a special but representative choice of the stability operators, a nonperturbative analysis demonstrates the existence of a critical density for the appearance of an overstability and the connected Hopf bifurcation, as suggested in a previous paper [Phys. Lett. A 180, 257 (1993)].
01 Jan 1996
TL;DR: A brief summary of energy methods for linear stability in dissipative magnetohydrodynamics is given in this article, where the methods are equally efficient for fixed and free boundary problems.
Abstract: A brief summary of energy methods for linear stability in dissipative magnetohydrodynamics is given. In this case, the methods are equally efficient for fixed and free boundary problems. Linear asymptotic stability has implications in nonlinear stability, at least for a modest but finite level of perturbations.
Cites background from "A manifest Hopf bifurcation in resi..."
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Book•
10 Sep 1993
TL;DR: In this article, the authors give bounds on the number of degrees of freedom and the dimension of attractors of some physical systems, including inertial manifolds and slow manifolds.
Abstract: Contents: General results and concepts on invariant sets and attractors.- Elements of functional analysis.- Attractors of the dissipative evolution equation of the first order in time: reaction-diffusion equations.- Fluid mechanics and pattern formation equations.- Attractors of dissipative wave equations.- Lyapunov exponents and dimensions of attractors.- Explicit bounds on the number of degrees of freedom and the dimension of attractors of some physical systems.- Non-well-posed problems, unstable manifolds. lyapunov functions, and lower bounds on dimensions.- The cone and squeezing properties.- Inertial manifolds.- New chapters: Inertial manifolds and slow manifolds the nonselfadjoint case.
5,038 citations
Book•
17 Aug 1976TL;DR: The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value as mentioned in this paper.
Abstract: The goal of these notes is to give a reasonably complete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to specific problems, including stability calculations Historically, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930 Hopf's basic paper [1] appeared in 1942 Although the term "Poincare-Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it Hopf's crucial contribution was the extension from two dimensions to higher dimensions
The principal technique employed in the body of the text is that of invariant manifolds The method of Ruelle-Takens [1] is followed, with details, examples and proofs added Several parts of the exposition in the main text come from papers of P Chernoff, J Dorroh, O Lanford and F Weissler to whom we are grateful
The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations; see for example, Hale [1,2] and Hartman [1] Of course, other methods are also available In an attempt to keep the picture balanced, we have included samples of alternative approaches Specifically, we have included a translation (by L Howard and N Kopell) of Hopf's original (and generally unavailable) paper These original methods, using power series and scaling are used in fluid mechanics by, amongst many others, Joseph and Sattinger [1]; two sections on these ideas from papers of Iooss [1-6] and Kirchgassner and Kielhoffer [1] (contributed by G Childs and O Ruiz) are given
The contributions of S Smale, J Guckenheimer and G Oster indicate applications to the biological sciences and that of D Schmidt to Hamiltonian systems For other applications and related topics, we refer to the monographs of Andronov and Chaiken [1], Minorsky [1] and Thom [1]
The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value In Hopf's original approach, the determination of the stability of the resulting periodic orbits is, in concrete problems, an unpleasant calculation We have given explicit algorithms for this calculation which are easy to apply in examples (See Section 4, and Section 5A for comparison with Hopf's formulae) The method of averaging, exposed here by S Chow and J Mallet-Paret in Section 4C gives another method of determining this stability, and seems to be especially useful for the next bifurcation to invariant tori where the only recourse may be to numerical methods, since the periodic orbit is not normally known explicitly
In applications to partial differential equations, the key assumption is that the semi-flow defined by the equations be smooth in all variables for t > O This enables the invariant manifold machinery, and hence the bifurcation theorems to go through (Marsden [2]) To aid in determining smoothness in examples we have presented parts of the results of Dorroh-Marsden [1] Similar ideas for utilizing smoothness have been introduced independently by other authors, such as D Henry [1]
Some further directions of research and generalization are given in papers of Jost and Zehnder [1], Takens [1, 2], Crandall-Rabinowitz [1, 2], Arnold [2], and Kopell-Howard [1-6] to mention just a few that are noted but are not discussed in any detail here We have selected results of Chafee [1] and Ruelle [3] (the latter is exposed here by S Schecter) to indicate some generalizations that are possible
The subject is by no means closed Applications to instabilities in biology (see, eg Zeeman [2], Gurel [1-12] and Section 10, 11); engineering (for example, spontaneous "flutter" or oscillations in structural, electrical, nuclear or other engineering systems; cf Aronson [1], Ziegler [1] and Knops and Wilkes [1]), and oscillations in the atmosphere and the earth's magnetic field (cf Durand [1]) are appearing at a rapid rate Also, the qualitative theory proposed by Ruelle-Takens [1] to describe turbulence is not yet well understood (see Section 9) In this direction, the papers of Newhouse and Peixoto [1] and Alexander and Yorke [1] seem to be important Stable oscillations in nonlinear waves may be another fruitful area for application; cf Whitham [1] We hope these notes provide some guidance to the field and will be useful to those who wish to study or apply these fascinating methods
After we completed our stability calculations we were happy to learn that others had found similar difficulty in applying Hopf's result as it had existed in the literature to concrete examples in dimension ≥ 3 They have developed similar formulae to deal with the problem; cf Hsu and Kazarinoff [1, 2] and Poore [1]
The other main new result here is our proof of the validity of the Hopf bifurcation theory for nonlinear partial differential equations of parabolic type The new proof, relying on invariant manifold theory, is considerably simpler than existing proofs and should be useful in a variety of situations involving bifurcation theory for evolution equations
These notes originated in a seminar given at Berkeley in 1973-4 We wish to thank those who contributed to this volume and wish to apologize in advance for the many important contributions to the field which are not discussed here; those we are aware of are listed in the bibliography which is, admittedly, not exhaustive Many other references are contained in the lengthy bibliography in Cesari [1] We also thank those who have taken an interest in the notes and have contributed valuable comments These include R Abraham, D Aronson, A Chorin, M Crandall, R Cushman, C Desoer, A Fischer, L Glass, J M Greenberg, O Gurel, J Hale, B Hassard, S Hastings, M Hirsch, E Hopf, N D Kazarinoff, J P LaSalle, A Mees, C Pugh, D Ruelle, F Takens, Y Wan and A Weinstein Special thanks go to J A Yorke for informing us of the material in Section 3C and to both he and D Ruelle for pointing out the example of the Lorentz equations (See Example 4B8) Finally, we thank Barbara Komatsu and Jody Anderson for the beautiful job they did in typing the manuscript
Jerrold Marsden
Marjorie McCracken
1,878 citations
TL;DR: In this article, a priori estimates for the vorticity of solutions of the Navier-Stokes equations were presented, which imply that the L 1 norm of the vortexicity is a priora bounded in time and that the time average of the 4/(3+e) power of the L 4/( 3+e)-power of the gradient of the VV is a posteriori bounded.
Abstract: We present new a priori estimates for the vorticity of solutions of the three dimensional Navier-Stokes equations. These estimates imply that theL
1 norm of the vorticity is a priori bounded in time and that the time average of the 4/(3+e) power of theL
4/(3+e) spatial norm of the gradient of the vorticity is a priori bounded. Using these bounds we construct global Leray weak solutions of the Navier-Stokes equations which satisfy these inequalities. In particular it follows that vortex sheet, vortex line and even more general vortex structures with arbitrarily large vortex strengths are initial data which give rise to global weak solutions of this type of the Navier-Stokes equations. Next we apply these inequalities in conjunction with geometric measure theoretical arguments to study the two dimensional Hausdorff measure of level sets of the vorticity magnitude. We obtain a priori bounds on an average such measure, . When expressed in terms of the Reynolds number and the Kolmogorov dissipation length η, these bounds are
$$\left\langle \mu \right\rangle \leqq \frac{{L^3 }}{\eta }\left( {1 + \operatorname{Re} ^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-
ulldelimiterspace} 2}} } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-
ulldelimiterspace} 2}} ,$$
The right-hand side of this inequality has a simple geometrical interpretation: it represents the area of a union of non-overlapping spheres of radii η which fill a fraction of the spatial domain. As the Reynolds number increases, this fraction decreases. We study also the area of level sets of scalars and in particular isotherms in Rayleigh-Benard convection. We define a quantity, 〈μ〉
r, t
(x
0) describing an average value of the area of a portion of a level set contained in a small ball of radiusr about the pointx
0. We obtain the inequality
$$\left\langle \mu \right\rangle _{r,t} \left( {x_0 } \right) \leqq C_\kappa ,^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-
ulldelimiterspace} 2}_r {5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-
ulldelimiterspace} 2}} \left\langle {v\left( {x_0 } \right)} \right\rangle ^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-
ulldelimiterspace} 2}} ,$$
wherek is the diffusivity constant and is the velocity. The inequality is valid forr small but larger than a small scale λ=k
−1. In the case of turbulent velocities this scale is smaller than the smallest significant physical scale, suggesting that 2.5 is a lower bound for the fractal dimension of an (ensemble) average interface in turbulent flow, a fact which agrees with the experimental lower bound of 2.35. We define a similar quantity 〈μ〉
δ, t
representing an average value of the area of a portion of a level set contained in the region of spaceD
δ={x∈D; dist (x, δD)>δ}, whereD is the domain of aspect ratio of order one and diameterL, where the convection takes place. We obtain the inequality
$$\left\langle \mu \right\rangle _{\delta ,t} \leqq C\left( {L^3 \delta ^{ - 1} + L^{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-
ulldelimiterspace} 2}} \delta ^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-
ulldelimiterspace} 2}} Ra^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-
ulldelimiterspace} 2}} } \right),$$
where Ra is the Rayleigh number.
112 citations
TL;DR: In this article, physically motivated test functions are introduced to simplify the stability functional, which makes its evaluation and minimization more tractable, and the simplified functional reduces to a good approximation of the exact stability functional.
8 citations
TL;DR: In this article, a sufficient stability condition with respect to purely growing modes is derived for resistive MHDs, which is a necessary and sufficient condition for purely growing MHD.
7 citations