By Gorban A.N.

The paper supplies a scientific research of singularities of transition tactics in dynamical structures. common dynamical structures with dependence on parameter are studied. A process of rest instances is developed. each one rest time will depend on 3 variables: preliminary stipulations, parameters $k$ of the method and accuracy $\epsilon$ of the relief. The singularities of leisure instances as features of $(x_0,k)$ less than mounted $\epsilon$ are studied. The type of alternative bifurcations (explosions) of restrict units is played. The kin among the singularities of rest instances and bifurcations of restrict units are studied. The peculiarities of dynamics which entail singularities of transition approaches with no bifurcations are defined besides. The analogue of the Smale order for normal dynamical platforms less than perturbations is built. it's proven that the perturbations simplify the placement: the interrelations among the singularities of rest instances and different peculiarities of dynamics for normal dynamical method less than small perturbations are just like for the Morse-Smale platforms.

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**singularities of transition processes in dynamical systems: qualitative theory of critical delays**

The paper offers a scientific research of singularities of transition techniques in dynamical structures. common dynamical structures with dependence on parameter are studied. A process of rest occasions is developed. each one leisure time depends upon 3 variables: preliminary stipulations, parameters $k$ of the method and accuracy $\epsilon$ of the comfort.

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**Extra info for singularities of transition processes in dynamical systems: qualitative theory of critical delays**

**Example text**

Let us choose for every i = 1, . . , n such Ti > T that f δ/2 (Ti |xi , k) = xi . Construct a periodical (k, y1 , ε)n motion passing through the points x1 , . . , xn with the period T0 = 2 t=1 Ti − T1 − Tn : let 0 ≤ t ≤ T0 , suppose f ∗ (t) f δ/2 (tx1 , k), j−1 δ/2 t − i=1 Ti |xi , k , f = n n−1 f δ/2 t − i=1 Ti + i=j+1 Ti |xj , k , if 0 ≤ t < T ; if j−1 i=1 Ti ≤ t < j i=1 Ti (j = 2, . . , n); if n−1 i=1 ≤t< n i=j+1 Ti n n−1 i=1 Ti + i=j Ti . Ti + If mT0 ≤ t < (m + 1)T0 , then f ∗ (t) = f ∗ (t − mT0 ).

Ti+1 − ti > T ). 38 A. N. GORBAN EJDE-2004/MON. 05 y is the ω-limit point of f ∗ (t), therefore, y ∈ ω ε+γ (x, k). 10. For any x ∈ X, k ∈ K the set ω0 (x, k) is closed and kinvariant. Proof. 9 it follows ω 0 (x, k) = ω ε (x, k). 6) ε>0 Therefore, ω 0 (x, k) is closed. Let us prove that it is k-invariant. Let y ∈ ω 0 (x, k). Then there are such sequences εj > 0, εj → 0, tji → ∞ as i → ∞ (j = 1, 2, . . ) and such family of (k, x, εj )-motions f εj (t|x, k) that f εj (tji |x, k) → y as i → ∞ for any j = 1, 2, .

E. the fact that every trajectory from ωF is a fixed point or a limit cycle and also from the fact that rough two-dimensional systems have no loops we conclude that τ1 -slow relaxations do exist. Thus, if ωF X is connected, then F |X has not even τ3 -slow relaxations, and if ωF X is disconnected, then there are η3 and τ1,2,3 -slow relaxations. 12 is proved. 12 is not always true. Really, let us consider topologically transitive U -flow F over the manifold M [5]; ωF = M , therefore η3 (x, ε) = 0 for any X ∈ M, ε > 0.