Abstract
Fundamentals of group-theoretic bifurcation theory are introduced as a mathematical methodology to deal with agglomeration in economic geography, which is captured in this book as bifurcation phenomena of systems with dihedral or hexagonal symmetry. The framework of group-theoretic bifurcation analysis of economic agglomeration is illustrated for the two-place economy in new economic geography. Symmetry of a system is described by means of a group, and group representation is introduced as the main tool used to formulate the symmetry of the equilibrium equation of symmetric systems. Group-theoretic bifurcation analysis procedure under group symmetry is presented with particular emphasis on Liapunov–Schmidt reduction under symmetry. Bifurcation equation, equivariant branching lemma, and block-diagonalization are introduced as mathematical tools used to tackle bifurcation of a symmetric system in Chaps. 3–9.
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- 1.
In fluid mechanics, for example, the Couette–Taylor flow in a hollow cylinder, which is a rotating annular fluid, displays wave patterns with various symmetries through pattern selection (e.g., Taylor, 1923 [19]). The convective motion of fluid in the Bénard problem displays regularly arrayed hexagons (e.g., Koschmieder, 1974 [11]). These phenomena can be explained successfully in terms of symmetry-breaking bifurcations.
- 2.
See, for example, Sattinger, 1979 [17], 1980; Chow and Hale, 1982 [2]; Golubitsky and Schaeffer, 1985 [6]; Golubitsky, Stewart, and Schaeffer, 1988 [7]; Mitropolsky and Lopatin, 1988 [14]; Allgower, Böhmer, and Golubitsky, 1992 [1]; Marsden and Ratiu, 1999 [12]; Olver, 1995 [15]; and Hoyle, 2006 [9].
- 3.
We restrict ourselves to finite-dimensional equations and finite groups, which are sufficient for our purpose. In so doing we can avoid mathematical sophistications necessary to address differential equations and continuous groups.
- 4.
By adding \(\lambda _{1}\omega _{1} =\lambda _{1}\overline{\omega }\) and \(\lambda _{2}\omega _{2} =\lambda _{2}\overline{\omega }\), we obtain \(\overline{\omega } =\lambda _{1}\omega _{1} +\lambda _{2}\omega _{2} = (\lambda _{1} +\lambda _{2})\overline{\omega }\), which implies \(\lambda _{1} +\lambda _{2} = 1\) since \(\overline{\omega } > 0\).
- 5.
- 6.
In new economic geography (Fujita, Krugman, and Venables, 1999 [5]), the value τ = τ A is called the break point and the bifurcation at point A is called tomahawk bifurcation.
- 7.
The value τ = τ C is an important index of the sustainability of the core–periphery pattern called the sustain point.
- 8.
- 9.
- 10.
- 11.
Loosely speaking, the term “generically” might be replaced by “unless the parameters take special values.”
- 12.
- 13.
The reduction procedure is valid for a critical point \((\boldsymbol{u}_{\mathrm{c}},f_{\mathrm{c}})\) that is not necessarily group-theoretic although in later chapters we always treat group-theoretic critical points.
- 14.
- 15.
U = (ker(J c)) ⊥ and V = (range(J c)) ⊥ in Remark 2.9 are valid choices, since T is assumed to be unitary (cf., (2.33)).
- 16.
We can replace \(T(g)\boldsymbol{w}\) by \(\boldsymbol{w}\) since \(\boldsymbol{w}\) is arbitrary in ker(J c) and ker(J c) is G-invariant.
- 17.
The uniqueness assertion applies since \(T(g)\boldsymbol{\varphi }(T({g}^{-1})\boldsymbol{w},\tilde{f}) \in U\) by the G-invariance of U.
- 18.
Proof of (2.130): \(h \in \varSigma (T(g)\boldsymbol{u})\;\Longleftrightarrow\;T(h)T(g)\boldsymbol{u} = T(g)\boldsymbol{u}\;\Longleftrightarrow\;T{(g)}^{-1}T(h)T(g)\boldsymbol{u} = \boldsymbol{u}\;\Longleftrightarrow\;T({g}^{-1}hg)\boldsymbol{u} = \boldsymbol{u}\;\Longleftrightarrow\;{g}^{-1}hg \in \varSigma (\boldsymbol{u})\;\Longleftrightarrow\;h \in g \cdot \varSigma (\boldsymbol{u}) \cdot {g}^{-1}\).
- 19.
See Proposition 2.1 (Schur’s lemma) in Sect. 2.3.4.
References
Allgower E, Böhmer K, Golubitsky M (eds) (1992) Bifurcation and symmetry. International series of numerical mathematics, vol 104. Birkhäuser, Basel
Chow S, Hale JK (1982) Methods of bifurcation theory. Grundlehren der mathematischen Wissenschaften, vol 251. Springer, New York
Cicogna G (1981) Symmetry breakdown from bifurcations. Lettere al Nuovo Cimento 31: 600–602
Curtis CW, Reiner I (1962) Representation theory of finite groups and associative algebras. Wiley classic library, vol 45. Wiley (Interscience), New York
Fujita M, Krugman P, Venables AJ (1999) The spatial economy: cities, regions, and international trade. MIT, Cambridge
Golubitsky M, Schaeffer DG (1985) Singularities and groups in bifurcation theory, vol 1. Applied mathematical sciences, vol 51. Springer, New York
Golubitsky M, Stewart I, Schaeffer DG (1988) Singularities and groups in bifurcation theory, vol 2. Applied mathematical sciences, vol 69. Springer, New York
Hamermesh M (1962) Group theory and its application to physical problems. Addison-Wesley series in physics. Addison-Wesley, Reading
Hoyle R (2006) Pattern formation: an introduction to methods. Cambridge texts in applied mathematics. Cambridge University Press, Cambridge
Ikeda K, Murota K (2010) Imperfect bifurcation in structures and materials: engineering use of group-theoretic bifurcation theory, 2nd edn. Applied mathematical sciences, vol 149. Springer, New York
Koschmieder EL (1974) Benard convection. Adv Chem Phys 26:177–188
Marsden JE, Ratiu TS (1999) Introduction to mechanics and symmetry, 2nd edn. Texts in applied mathematics, vol 17. Springer, New York
Miller W Jr (1972) Symmetry groups and their applications. Pure and applied mathematics, vol 50. Academic, New York
Mitropolsky YuA, Lopatin AK (1988) Nonlinear mechanics, groups and symmetry. Mathematics and its applications. Kluwer, Dordrecht
Olver PJ (1995) Equivariance, invariants, and symmetry. Cambridge University Press, Cambridge
Pflüger M (2004) A simple, analytically solvable, Chamberlinian agglomeration model. Reg Sci Urban Econ 34(5):565–573
Sattinger DH (1979) Group theoretic methods in bifurcation theory. Lecture notes in mathematics, vol 762. Springer, Berlin
Serre J-P (1977) Linear representations of finite groups. Graduate texts in mathematics, vol 42. Springer, New York
Taylor GI (1923) The stability of a viscous fluid contained between two rotating cylinders. Phil Trans Roy Soc Lond Ser A 223:289–343
Vanderbauwhede A (1980) Local bifurcation and symmetry. Habilitation thesis, Rijksuniversiteit Gent
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Ikeda, K., Murota, K. (2014). Group-Theoretic Bifurcation Theory. In: Bifurcation Theory for Hexagonal Agglomeration in Economic Geography. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54258-2_2
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