Bifurcation and symmetry breaking in applied mathematics

Sattinger, D. H.

doi:10.1090/s0273-0979-1980-14823-5

Cited by 111 publications

(26 citation statements)

References 70 publications

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“…Namely, by inspecting the second γ-derivative of U s (γ) at γ = π/3 one finds that the universal equilibrium family s → γ * = π/3 yields a relative minimum of U s (γ) for s > −4 and a relative maximum for s < −4, while the second γ-derivative of U s (γ) at γ = π/3 vanishes at s = −4. Since (s, γ) → U s (γ) is real analytic about (s, γ) = (−4, π/3), analytical bifurcation theory [Sat80] reveals that at s = −4 two continuous families of non-universal equilibria branch off of the universal equilateral one. In a neighborhood of s = −4 it can be computed by setting s = −4 + 2σ, |σ| 1, and ξ 2 * (s) = 1 + η(σ), with 0 < |η(σ)| 1, and Taylor-expanding (19).…”

Section: Proof Of Propositionmentioning

confidence: 99%

Hilbert’s ‘Monkey Saddle’ and Other Curiosities in the Equilibrium Problem of Three Point Particles on a Circle for Repulsive Power Law Forces

Kiessling

2019

J Dyn Diff Equat

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This article determines all possible (proper as well as pseudo) equilibrium arrangements under a repulsive power law force of three point particles on the unit circle. These are the critical points of the sum over the three (standardized) Riesz pair interaction terms, each given by V s (r) = s −1 (r −s − 1) when the real parameter s = 0, and by V 0 (r) := lim s→0 V s (r) = − ln r; here, r is the chordal distance between the particles in the pair. The bifurcation diagram which exhibits all these equilibrium arrangements together as functions of s features three obvious "universal" equilibria, which do not depend on s, and two not-so-obvious continuous families of s-dependent non-universal isosceles triangular equilibria. The two continuous families of non-universal equilibria are disconnected, yet they bifurcate off of a common universal limiting equilibrium (the equilateral triangular configuration), at s = −4, where the graph of the total Riesz energy of the 3-particle configurations has the shape of a "monkey saddle." In addition, one of the families of non-universal equilibria also bifurcates off of another universal equilibrium (the antipodal arrangement), at s = −2. While the bifurcation at s = −4 is analytical, the one at s = −2 is not. The bifurcation analysis presented here is intended to serve as template for the treatment of similar N -point equilibrium problems on S d for small N .

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Section: Proof Of Propositionmentioning

confidence: 99%

Hilbert’s ‘Monkey Saddle’ and Other Curiosities in the Equilibrium Problem of Three Point Particles on a Circle for Repulsive Power Law Forces

Kiessling

2019

J Dyn Diff Equat

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“…In principle -and forgetting about the delays -the underlying analysis has been done by , who took advantage of Sattinger's (1979Sattinger's ( , 1980Sattinger's ( , 1983 beautiful work on bifurcation at an eigenvalue whose (infinite) degeneracy stems from a spatial symmetry group. The only proviso of their analysis is that, roughly, the inhibition has so long a range that the patterns turn out to be stationary parallel stripes; in more recent terminology (Murray 1989), they find Turing patterns.…”

Section: Symmetriesmentioning

confidence: 99%

Continuum limit of discrete neuronal structures: is cortical tissue an “excitable” medium?

Hemmen

2004

Biol Cybern

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As a simple model of cortical tissue, we study a locally connected network of spiking neurons in the continuum limit of space and time. This is to be contrasted with the usual numerical simulations that discretize both of them. Refractoriness, noise, axonal delays, and the time course of excitatory and inhibitory postsynaptic potentials have been taken into account explicitly. We pose, and answer, the question of whether the continuum limit presents a full description of scenarios found numerically (the answer is no, not quite). In other words, can the numerics be reduced to a continuum description of a well-known type? As a corollary, we derive some classical results such as those of Wilson and Cowan (1973), thus indicating under what conditions they are valid. Furthermore, we show that spatially discrete objects may be fragile due to noise arising from the stochasticity of the individual neurons, whereas they are not once the continuum limit has been taken. This, then, resolves the above question. Finally, we indicate how one can directly incorporate orientation preference of the neurons.

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“…On the theoretical side, there is an extensive literature on the analysis of bifurcation and symmetry breaking by using the group-representation theory, e.g. Sattinger (1980).…”

Section: Introductionmentioning

confidence: 99%

Investigation of symmetry breaking in periodic gravity–capillary waves

2016

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In this paper, fully nonlinear non-symmetric periodic gravity-capillary waves propagating at the surface of an inviscid and incompressible fluid are investigated. This problem was pioneered analytically by Zufiria (J. Fluid Mech., vol. 184, 1987c, pp. 183-206) and numerically by Shimizu & Shōji (Japan J. Ind. Appl. Maths, vol. 29 (2), 2012, pp. 331-353). We use a numerical method based on conformal mapping and series truncation to search for new solutions other than those shown in Zufiria (1987c) and Shimizu & Shōji (2012). It is found that, in the case of infinite-depth, non-symmetric waves with two to seven peaks within one wavelength exist and they all appear via symmetry-breaking bifurcations. Fully exploring these waves by changing the parameters yields the discovery of new types of non-symmetric solutions which form isolated branches without symmetry-breaking points. The existence of non-symmetric waves in water of finite depth is also confirmed, by using the value of the streamfunction at the bottom as the continuation parameter.

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Bifurcation and symmetry breaking in applied mathematics

Cited by 111 publications

References 70 publications

Hilbert’s ‘Monkey Saddle’ and Other Curiosities in the Equilibrium Problem of Three Point Particles on a Circle for Repulsive Power Law Forces

Hilbert’s ‘Monkey Saddle’ and Other Curiosities in the Equilibrium Problem of Three Point Particles on a Circle for Repulsive Power Law Forces

Continuum limit of discrete neuronal structures: is cortical tissue an “excitable” medium?

Investigation of symmetry breaking in periodic gravity–capillary waves

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