Now let U be an

invariant subspace with basis U, and let U = U + [DELTA]U with a small perturbation [DELTA]U.

Hasan, "Rational

invariant subspace approximations with applications," IEEE Transactions on Signal Processing, vol.

Furthermore, authors have shown that CLBS is closely related to the

invariant subspace; namely, exact solutions defined on

invariant subspaces for equations or their variant forms can be obtained by using the CLBS method [12-24].

The subspace [K.sup.2.sub.u] = [H.sup.2] [member of] u[H.sup.2] is a proper nontrivial

invariant subspace of [S.sup.*], the most general one by the well-known theorem of A.

Furthermore, for the nonlinear problem, the multiple exp-function method [18, 19], the transformed rational function method [20-22], and

invariant subspace method [23, 24] are three systematical approaches to handle the nonlinear terms.

The operator A is polaroid if it is polar at every [lambda] [member of] iso[sigma](A), and it is hereditarily polaroid if every restriction [A|.sub.M] of A to an (always closed)

invariant subspace M of A is polaroid.

According to (28) and (32), the three-dimensional

invariant subspace is obtained as

In fact, the [S.sub.[infinity]]-

invariant subspace C{[[summation].sub.i[member of]N] [[upsilon].sub.i]} does not have a closed [CS.sub.[infinity]]-invariant complement, so that all projection operators with range C{[[summation].sub.i[member of]N] [[upsilon].sub.i]} are unbounded.

The

invariant subspace method is refined to present more unity and more diversity of exact solutions by taking subspaces of solutions to linear ordinary differential equations as

invariant subspaces that evolution equations admit [6].

(v) If M is an

invariant subspace of T, then the restriction has [T|.sub.M] has property (i).

A subspace M is invariant for T if T(M) [subset not equal to] M and a part of an operator is a restriction of it to an

invariant subspace.

To be more precise, we consider two coupled map families such that the family maps all have the same fixed point which is nested within the same topologically transitive invariant set which is nested in turn within the same

invariant subspace. We prove in such a case that these point bifurcations which are transversal to the

invariant subspace generate two periodic of period 2 points in a neighbourhood of the given point and besides can simultaneously give rise to orbits that are homoclinic to the periodic points.