Theory ODE

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• Definition 3.1 Weak solution A continuous function u:(t+, t-) ->R is said to be a weak solution if it satisfies t-intt+ v'(t)u(t)+v(t)u'(t) dt = 0 for all v € Cc' ((t-, t+))
• Definition 4.1 Fixed Point The point q € Rn such that F(q) = 0 are called fixed points (or critical points och equilibrium points) of the dynamical system u' = F(u)
• Definition 4.2 Linearization The linear dynamical system u'=Au, A = nablaF(q) is called the linearization of the dynamical system u'=F(u) around the point q
• Definition 4.3 Stability of fixed points The fixed point q€Rn of the dynamical system u'=F(u) is said to be stable if for all e > 0 there exists a delta=delta(e) > 0 such that for all u0€Rn satisfying ||u0-q|| < delta, the solution u(t) with initial data u(0) = u0 is defined for all t≥0. A fixed point that is not stable is said to be unstable. If q is a stable fixed point and in addition there exists r>0 such that u(t) -> q as t-> inf, for all u0€Rn satisfying ||u0 -q||<r, then q is said to be locally asymptotically stable. If u(t) ->q as t->inf for all u0€Rn, then q is said to be globally asymptotically stable.
• Definition 4.4 Hamiltonian function A 2-dimensional dynamical system is said to be hamiltonian if there exists a function H:R2->R such that u1'=∂H/∂u2, u2'=∂H/∂u1, The function H is called hamiltonian function of the dynamical system
• Definition 4.5 names of fixed points the fixed point q of the dynamical system u'=F(u), F:Rn->Rn is said to be hyperbolic if all eigenvalues of the matrix A = lambdaF(q) have non-zero real parts. If all eigenvalues of A have negative real parts, then q is called a sink. If the eigenvalues of A have all positive real parts, then q is called a source. If at least one eigenvalue of A has positive real part, and at least one has negative real part, then q is called a saddle.
• Definition 5.1 convergence and stability of the finite difference method Let Q={t0, t1, ..., tn =T} be the partition of the interval [t0, T] with size h>0. Assume that u'=F(t, u), u(t0) = u0, has a unique strong solution in the interval [t0, T], with initial data u(t0) = u0. Let {u0, u1, ..., un] be an approximate solution on the partition Q of this initial value problem obtained by some numerical method (eg any finite difference method). The numerical method is said to be of order m if there exists a constant C independent of h such that ||u(T)-un|| ≤ Ch^m
• Definition 7.1 Weak/variational formulation of the boundary value problem The problem: Find u€C0'([0, X]) such that -0∫X(v'(x)u'(x))dx=0∫X(v(x)f(x))dx, for all v€C0'([0, X]), is called the weak/variational formulation of the boundary value probelm in the space C0'([0, X])
• Definition 3.1 Weak solution A continuous function u:(t+, t-) ->R is said to be a weak solution if it satisfies t-intt+ v'(t)u(t)+v(t)u'(t) dt = 0 for all v € Cc' ((t-, t+))

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