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His main focus is on attractors and various other invariant sets of such maps, and the related bifurcation structures observed in their parameter space.
In Section 2 we present the Schwarzian derivative {z, x} and the invariant I(z) of the differential equation [u.sub.zz] + f(z)[u.sub.z] + g(z)u = 0 through acting the similarity transformation [psi](z) = [phi](z)u(z) on the Schrodinger equation.
About algebraic invariant curves, as far as we know, there are few papers to consider switching system with algebraic invariant curves.
From the invariant property and the power series method [21], we have the expression
Furthermore, it is known that the Reidemeister torsion is an invariant under subdivision of the cell decomposition of W with p-coefficients up to factor [+ or -]1.
Every generalized Wallach space admits a three parameter family of invariant Riemannian metrics determined by Ad( K) -invariant inner products
With the help of the Ergodic Decomposition Theorem (see Proposition 1) it is possible to show that there exists a set N [member of] F with P(N) = 0 where, for any [omega] [not member of] N, [P.sup.[theta].sub.n]([omega], x) are probability measures that converge weakly, and the limiting distribution is an ergodic invariant probability measure Q([omega], x) on F.
[Dev.sup.i,] = 0, if [Dev.sup.i,] < [T.sub.AC] and this establishes an affine invariant hypothesis.
The proposed illumination invariant extraction method consists of three steps.
If u and v are invariants of a Lie algebra admitted by any system of ordinary differential equations, then dv/du is also its invariant.
If f is an endomorphism of M, then H(x) [less than or equal to] H(f(x)), and therefore L is fully invariant. Therefore with every large submodule L of M we may associate a sequence n(L).
The conditions given in [12] were developed for discrete-time nonlinear systems and depend on the existence of a certain controlled invariant vector function.