User:Styagi99/sandbox

Chafee Theorem
Consider an autonomous differential equation

x' = P( ԑ ) + X ( x, ԑ )                                                               ...(*)

where x, X Є $$R^n$$(n > 2), ԑ > 0 is a small parameter and A is a real n x n matrix. =Assumptions= (A1) There exist numbers r0> 0 and ԑ0 > 0 such that A is continuous on [0,ԑ0] and X is continuous on the domain $$ B^n$$(r) × [0,ԑ0] •

(A2) For each in [O,ԑ0] we have X(O,ԑ) = 0 so that the origin x = 0 is an equilibrium point of (*).ԑ

(A3) For each r in [O,r0] there exists a k(r) > 0 such that on the domain $$B^n$$(r) × [O,ԑ0] the function X is uniformly Lipschitzian in x with Lipschitz constant k(r); moreover, k(r) → 0 as r → O.


 * X(x1,ԑ)-X(x2,ԑ)| ≤k(r) ||x1-x2||,

(x1,x2 є  B(r),ԑ  є  [0,ԑ0]).

(A4) For each ԑ in [O,ԑ0] the matrix A(ԑ) has a complex-conjugate pair of eigenvalues a(ԑ) ± ib(ԑ) whose real and imaginary parts satisfy the conditions

b(ԑ) > 0 a(O) = 0, a(ԑ) > 0

The other eigen values λ1(ԑ), λ2(ԑ),•.., λn-2(ԑ) of A(ԑ)have their real parts negative for all ԑ in [O,ԑ0].

(A5) For ԑ= 0 the equilibrium point of (*) at the origin is asymptotically stable in the sense of Liapunov.

Hypotheses (A1) and (A3) are sufficient to guarantee the usual properties of existence, uniqueness, and continuity in initial conditions for solutions of (*)