By Walter E Thirring

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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**Additional resources for A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems**

**Example text**

A detailed analysis of such situations is given in [MR80] and [MKKR94]. 31) for x, y ∈ R , again under the assumptions f (0, 0) = ∂x f (0, 0) = 0 and g(0, 0) > 0 . 32) In this section we assume that f and g are of class C 3 . If, unlike in the case of the saddle–node bifurcation, ∂xx f (0, 0) vanishes, several new kinds of bifurcations can occur. These bifurcations are not generic, unless f and g are restricted to belong to some smaller function space, for instance functions satisfying certain symmetry conditions.

4. 1) that the right-hand side does not explicitly depend on ε. This is no real constraint as we can always introduce a dummy variable for ε. 10) as and consider z as an additional slow variable. 2 2 Some authors do not allow for ε-dependent slow manifolds and consider x (y, 0), obtained by setting ε = 0, as the slow manifold. 1 Slow Manifolds 21 The following example provides an application with ε-dependent righthand side. 5 (Stommel’s box model). Simple climate models, whose dynamic variables are averaged values of physical quantities over some large volumes, or boxes, are called box models.

46) and the equilibrium branch. The dynamics of zt is governed by an equation of the form 6 This assumption is made to avoid situations such as f (x, y) = x5/2 − y, in which r = 2 but f20 (x, y) ≡ 0, where the relation between Newton’s polygon and slow manifold does not work. 38 2 Deterministic Slow–Fast Systems (a) (b) (c) k k k 2 2 1 p j p j p j 1 2 1 2 1 3 Fig. 7. Newton’s polygon for (a) the saddle–node, (b) the transcritical and (c) the asymmetric pitchfork bifurcation. The saddle–node bifurcation has branches of exponent q = 1/2, for which p = 1/2.