By Chao K.C., Robinson R.L. (eds.)

ISBN-10: 0841205000

ISBN-13: 9780841205000

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Additional resources for Equations of State in Engineering and Research

Sample text

Thus, y is a global solution of the RFDE(F ). On the other hand y(0) = ψ. Consequently, ω(ϕ) is invariant. The assertions for ω(S), S ⊂ M , which are contained in the statement, can now be easily proved and the assertions relative to α(ϕ), ϕ ∈ M , are proved in an analogous way. Given an RFDE(F ) on M , we denote by A(F ) the set of all initial data of global bounded solutions of F . The set A(F ) is clearly an invariant set of F . 1 RFDE on manifolds 25 implies that ω(ϕ) (or α(ϕ)) is contained in A(F ).

10 Let X, Λ be complete metric spaces, T : Λ × X → X continuous and suppose there is a bounded set B independent of λ ∈ Λ such that B is compact dissipative under T (λ, ·) for every λ ∈ Λ. If T is collectively β-contracting, then the maximal compact invariant set J(λ) of T (λ, ·) is upper semicontinuous in λ. For an historical discussion of the existence of maximal compact invariant sets, see Hale [72], [73]. The proofs of all results also can be found there. We remark that more sophisticated results on dissipative systems have been obtained by Massat [138].

Proof: One ﬁrst observes that the critical points in the equator are not hyperbolic if the eigenvalues of A are not distinct. This implies that A contains either matrices of types I or II. One then shows that the set of all real nonsingular 2 × 2 matrices A with distinct eigenvalues is open and dense in M (2). For matrices of type (I), the critical points are N, S and four points in the equator given in terms of the polar angle by θ = 0, π/2, 3π/2 and π. The hyperbolicity of N and S is equivalent to −a1 , −a2 = (π/2 + 2nπ), n = 0, ±1, ±2.