singularities of transition processes in dynamical systems: by Gorban A.N.

By Gorban A.N.

The paper offers a scientific research of singularities of transition techniques in dynamical platforms. basic dynamical platforms with dependence on parameter are studied. A procedure of rest instances is built. every one leisure time will depend on 3 variables: preliminary stipulations, parameters $k$ of the procedure and accuracy $\epsilon$ of the comfort. The singularities of leisure instances as capabilities of $(x_0,k)$ less than mounted $\epsilon$ are studied. The class of other bifurcations (explosions) of restrict units is played. The family among the singularities of leisure occasions and bifurcations of restrict units are studied. The peculiarities of dynamics which entail singularities of transition strategies with out bifurcations are defined to boot. The analogue of the Smale order for common dynamical platforms below perturbations is developed. it's proven that the perturbations simplify the location: the interrelations among the singularities of rest occasions and different peculiarities of dynamics for common dynamical process less than small perturbations are similar to for the Morse-Smale structures.

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10. Let t0 < 0. Find ε0 > 0 and δ0 > 0 such that ε˜(χ(ε0 , T − t0 , T ), t0 , K) + ε˜(δ0 , t0 − T, k) < γ/2. 11 this is possible. Let us take εj < ε0 and choose from the sequence tji (i = 1, 2, . . ) such monotone subsequence tl (l = 1, 2, . . ) that tl > −t0 , ρ(f εi (tl |x, k), y) < δ0 and tl+1 − tl > T − t0 . Suppose f ∗ (t) = f εj (t|x, k), if t ∈ [tl + t0 , tl ] for any l = 1, 2, . . ; f (t − tl , y, k), if t ∈ [tl + t0 , tl ] (l = 1, 2, . . ). where f ∗ (t) is (k, x, γ)-motion, and z is the ω-limit point of this motion.

That tl > −t0 , ρ(f εi (tl |x, k), y) < δ0 and tl+1 − tl > T − t0 . Suppose f ∗ (t) = f εj (t|x, k), if t ∈ [tl + t0 , tl ] for any l = 1, 2, . . ; f (t − tl , y, k), if t ∈ [tl + t0 , tl ] (l = 1, 2, . . ). where f ∗ (t) is (k, x, γ)-motion, and z is the ω-limit point of this motion. Thus, z ∈ ω γ (x, k) for any γ > 0. The proposition is proved. 12. Let x ∈ ω 0 (x, k). Then for any ε > 0 there exists periodical (k, x, ε)-motion (This is a version of C 0 -closing lemma). EJDE-2004/MON. 05 SINGULARITIES OF TRANSITION PROCESSES 39 Proof.

F εj (t|xj , kj ) be the εj motions, ρ(f εj (τj |xj , kj ), xj+1 ) < δj , ρK (kj , k ∗ ) < δj /2. Then the mapping f ∗ (t) = f ε0 (t|x0 , k0 ), f εj t− if 0 ≤ t < τ0 ; i−1 j=0 τj |xj , kj , if i−1 j=0 τj ≤t< i j=0 τj , is (k ∗ , x0 , β)-motion, where β = sup {εj+1 + ε(εj + δj + δj+1 + ε(δj , T ), T )}. 5 follows directly from the definitions. 6. Let xi ∈ X, ki ∈ K, ki → k ∗ , εi > 0, εi → 0, f εi (t|xi , ki ) be (ki , xi , εi )-motions, ti > 0, ti > t0 , f εi (ti |xi , ki ) → x∗ . Then (k ∗ , x∗ )-motion is defined over the segment [−t0 , ∞) and f εi (t0 +t|xi , ki ) tends to f (t, x∗ , k ∗ ) uniformly over any compact segment from [−t0 , ∞).

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