\sum^{q-1}_{j=1} t_j. \end{cases} $$ Here $f^*(t)$ is $(y_2, \varepsilon)$-motion, and $z_n = z$ is its $\omega$-limit point. The set of those $z \in \omega^0_f$ for which exist such chains $z_1, \dots, z_n\ (n=1, 2, \dots)$ is an openly-closed (in $\omega^0_f$) subset of $\omega^0_f$, saturated downwards. Supposing $0 < \varepsilon \leq \varepsilon_0$, we obtain the needed result. Even more strong statement was proved: $Y$ can be chosen openly-closed (in $\omega^0_f$), not only open. Let us return to the proof of Theorem \ref{T4.4}. Since $\omega = \bigcup^{\infty}_{i=1} \omega^0_f (x_i)$ and each $z \in \omega^0_f (x_i)$ has an open (in $\omega^0_f$) saturated downwards neighbourhood $W_z$ which does not contain $y^*$, then the union of these neighborhoods is an open (in $\omega^0_f$) saturated downwards set which includes $\omega$ but does not contain $y^*$. Denote this set by $W$: $W = \bigcup_{z \in \omega} W_z$. Since $x_i \in At^0 (W),\ x \not \in At^0 (W)$ and $x_i \to x$, then $x \in \partial At^0 (W)$. The theorem is proved. \end{proof} The following proposition will be used in Subsection \ref{SS4.4} when studying slow relaxations of one perturbed system. \begin{proposition}\label{P4.22} Let $X$ be connected, $\omega^0_f$ be disconnected. Then there is such $x \in X$ that $x$-motion is whole and $x \not \in \omega^0_f$. There is also such partition of $\omega^0_f$ into openly-closed (in $\omega^0_f$) subsets: $$ \omega^0_f = W_1 \cup W_2,\quad W \cap W_2 = \varnothing,\quad \alpha_f (x) \subset W_1 \quad \mbox{but}\quad \omega^0_f (x) \subset W_2. $$ \end{proposition} \begin{proof} Repeating the proof of Lemma \ref{L3.3} (the repetition is practically literal, $\omega^0_f$ should be substituted instead of $\overline{\omega_f}$), we obtain that $\omega^0_f$ is not Lyapunov stable. Then, according to Lemma \ref{L3.2}, there is such $x \in X$ that $x$-motion is whole and $x \not \in \omega^0_f$. Note now that the set $\alpha_f(x)$ lies in equivalence class by the relation $\sim$, and the set $\omega^0_f$ is saturated by the relation $\sim$ (Proposition \ref{P4.17}, Lemma \ref{L4.5}). $\alpha_f (x) \bigcap \omega^0_f (x) = \varnothing$, otherwise, according to Proposition \ref{P4.17} and Lemma \ref{L4.4}, $x \in \omega^0_f$. Since $\omega^0_f / \sim$ is totally disconnected space (Proposition \ref{P4.18}), there exists partition of it into openly-closed subsets, one of which contains image of $\alpha_f (x)$ and the other contains image of $\omega^0_f (x)$ (under natural projection $\omega^0_f \to \omega^0_f / \sim$). Prototypes of these openly-closed sets form the needed partition of $\omega^0_f$. The proposition is proved. \end{proof} \subsection{Slow Relaxations in One Perturbed System}\label{SS4.4} In this subsection, as in the preceding one, we investigate one semiflow of homeomorphisms $f$ over a compact space $X$. \begin{theorem}\label{T4.5} $\eta^0_1$- and $\eta_2^0$-slow relaxations are impossible for one semiflow. \end{theorem} \begin{proof} It is enough to show that $\eta^0_2$-slow relaxations are impossible. Suppose the contrary: there are such $\gamma>0$ and such sequences of numbers $\varepsilon_i > 0\ \varepsilon_i \to 0$, of points $x_i \in X$ and of $(x_i, \varepsilon_i)$-motions $f^{\varepsilon_i} (t | x_i)$ that $\eta^{\varepsilon_i}_2 (f^{\varepsilon_i} (t | x_i), \gamma) \to \infty$. Similarly to the proofs of the theorems \ref{T4.3} and \ref{T3.1}, find a subsequence in $\{ f^{\varepsilon_i} (t | x_i) \}$ and such $y^* \in X$ that $\rho^* (y^*, \omega^0_f) \geq \gamma$ and, whatever be the neighbourhood $V$ of the point $y^*$ in $X$, $\overline{\mathop{\rm meas}} \{ t \geq 0 : f^{\varepsilon_i} (t | x_i) \in V \} \to \infty\ (i \to \infty, f^{\varepsilon_i} (t | x_i)$ belongs to the chosen subsequence). As in the proof of Theorem \ref{T4.3}, we have $y^* \in \omega^0_f (y^*) \subset \omega^0_f$. But, according to the constructing, $\rho^* (y^*, \omega^0_f) \geq \gamma > 0$. The obtained contradiction proves the absence of $\eta^0_2$-slow relaxations. \end{proof} \begin{theorem}\label{T4.6} Let $X$ be connected. Then, if $\omega^0_f$ is connected then the semiflow $f$ has not $\tau^0_{1,2,3}$- and $\eta^0_3$-slow relaxations. If $\omega^0_f$ is disconnected, then $f$ possesses $\tau^0_{1,2,3}$- and $\eta^0_3$-slow relaxations. \end{theorem} \begin{proof} Let $X$ and $\omega^0_f$ be connected. Then, according to the propositions \ref{P4.17} and \ref{P4.18}, $\omega^0_f (x) = \omega^0_f$ for any $x \in X$. Consequently, $\omega^0 (x)$-bifurcations are absent. Therefore (Theorem \ref{T4.1}) $\tau_3$-slow relaxations are absent. Consequently, there are not other $\tau^0_i$- and $\eta^0_i$-slow relaxations due to the inequalities $\tau^{\varepsilon}_i \leq \tau^{\varepsilon}_3$ and $\eta^{\varepsilon}_i \leq \tau^{\varepsilon}_3\ (i = 1,2,3)$ 1,2,3) (see Proposition \ref{P4.16}). The first part of the theorem is proved. Suppose now that $X$ is connected and $\omega^0_f$ is disconnected. Let us use Proposition \ref{P4.22}. Find such $x \in X$ that $x$-motion is whole, $x \not \in \omega^0_f$, and such partition of $\omega^0_f$ into openly-closed subsets $\omega^0_f = W_1 \bigcup W_2,\ W_1 \bigcap W_2 = \varnothing$ that $\alpha_f (x) \subset W_1,\ \omega^0_f (x) \subset W_2$. Suppose $\gamma = \frac{1}{3} r (W_1, W_2)$. There is such $t_0$ that for $t < t_0\ \rho^* (f(t,x), W_2) > 2 \gamma$. Let $p \in \alpha_f (x),\ t_j < t_0,\ t_j \to - \infty,\ f(t_j, x) \to p$. For each $j = 1,2, \dots$ exists such $\delta_j > 0$ that for $\varepsilon < \delta_j\ d(\omega^{\varepsilon}_f (f (t_j, x)), \omega^0_f (f (t_i, x))) < \gamma$ (this follows from the Shura-Bura lemma and Proposition \ref{P4.8}). Since $\omega^0_f (f (t_j, x)) = \omega^0_f (x)$ (Corollary \ref{C4.5}), for $\varepsilon < \delta_j\ d(\omega^{\varepsilon}_f (f(t_j, x)), W_2) < \gamma$. Therefore $\rho^* (f (t,x), \omega^{\varepsilon}_f (f (t_j, x))) > \gamma$ if $t \in [t_j, t_0],\ \varepsilon > \delta_j$. Suppose $x_i = f(t_j,\ x)$, $\varepsilon_j > 0$, $\varepsilon_j < \delta_j$, $ \varepsilon_j \to 0$, $ f^{\varepsilon_j} (t | x_j) = f(t,x_j)$. Then $\tau^{\varepsilon_j}_1 (f^{\varepsilon_j} (t | x_j), \gamma) \geq t_0 - t_i \to \infty$. The existence of $\tau_1$- (and consequently of $\tau_{2,3}$-)-slow relaxations is proved. To prove the existence of $\eta_3$-slow relaxations we need the following lemma. \end{proof} \begin{lemma}\label{L4.7} For any $\varepsilon > 0$ and $\varkappa > 0$, $\overline{\omega^{\varepsilon}_f} \subset \omega^{\varepsilon+ \varkappa}_f$. \end{lemma} \begin{proof} Let $y \in \overline{\omega^{\varepsilon}_f}$: there are such sequences of points $x_i \in X,\ y_i \in \omega^{\varepsilon}_f$, of $(x_i, \varepsilon)$-motions $f^{\varepsilon} (t | x_i)$, of numbers $t^i_j > 0,\ t^i_j \to \infty \ \mbox{as} \ j \to \infty $ that $y_i \to y,\ f^{\varepsilon} (t^i_j | x_i) \to y_i \ \mbox{as} \ j \to \infty$. Suppose that $\delta = \frac{1}{2} \delta (\frac{\varkappa}{3}, T)$. There is such $y_i$ that $\rho (y_i, y) < \delta$. For this $y_i$ there is such monotone sequence $t_j \to \infty$ that $t_j - t_{j-1} > T$ and $\rho (y_i, f^{\varepsilon} (t_j | x_i )) < \delta$. Suppose $$ f^* (t) = \left\{ \begin{array}{ll} f^{\varepsilon} (t | x_i), & \mbox{if }t \neq t_j;\\ y, & \mbox{if }t = t_j\ (j = 1, 2, \dots), \end{array} \right. $$ where $f^* (t)$ is $(x_i, \varkappa + \varepsilon)$-motion and $y \in \omega (f^*)$. Consequently, $y \in \omega^{\varepsilon + \varkappa}_f$. The lemma is proved. \end{proof} \begin{corollary}\label{C4.7} $\omega^0_f = \bigcap_{\varepsilon > 0} \overline{\omega^{\varepsilon}_f}$. \end{corollary} Let us return to the proof of Theorem \ref{T4.6} and show the existence of $\eta^0_3$-slow relaxations if $X$ is connected and $\omega^0_f$ is not. Suppose that $\gamma = \frac{1}{5} r (W_1, W_2)$. Find such $\varepsilon_0 > 0$ that for $\varepsilon < \varepsilon_0\ d(\omega^{\varepsilon}_f, \omega^0_f) > \gamma$ (it exists according to Corollary \ref{C4.7} and the Shura-Bura lemma). There is $t_1$ for which $d(f(t_1, x), \omega^0_f) > 2 \gamma$. Let $t_j < t_1,\ t_j \to - \infty,\ x_j = f(x_j, x)$. As $(x_j, \varepsilon)$-motions let choose true motions $f(t,x_j)$. Suppose that $\varepsilon_j \to 0,\ 0 < \varepsilon_j < \varepsilon_0$. Then $\eta^{\varepsilon_j} (f(t,x_j), \gamma) > t_1 - t_j \to \infty$ and, consequently, $\eta^0_3$-slow relaxations exist. The theorem is proved. In conclusion of this subsection let us give {\it the proof of Theorem \ref{T4.2}.} We consider again the family of parameter depending semiflows. \begin{proof}[Proof of Lemma \ref{C4.7}] Let $X$ be connected and $\omega^0(x,k)$-bifurcations exist. Even if for one $k \in K\ \omega^0 (k)$ is disconnected, then, according to Theorem \ref{T4.6}, $\tau^0_3$-slow relaxations exist. Let $\omega^0 (k)$ be connected for any $k \in K$. Then $\omega^0 (x,k) = \omega^0 (k)$ for any $x \in X,\ k \in K$. Therefore from the existence of $\omega^0 (x,k)$-bifurcations follows in this case the existence of $\omega^0 (k)$-bifurcations. Thus, Theorem \ref{T4.2} follows from the following lemma which is of interest by itself. \end{proof} \begin{lemma}\label{L4.8} If the system (\ref{e1}) possesses $\omega^0 (k)$-bifurcations, then it possesses $\tau^0_3$- and $\eta^0_3$-slow relaxations. \end{lemma} \begin{proof} Let $k^*$ be a point of $\omega^0 (k)$-bifurcation: and there are such $\alpha > 0,\ y^* \in \omega^0 (k^*)$ that $\rho^* (y^*, \omega^0 (k_i)) > \alpha > 0$ for any $i = 1, 2, \dots$. According to Corollary \ref{C4.7} and the Shura-Bura lemma, for every $i$ exists $\delta_i > 0$ for which $\rho^*(y^*, \omega^{\delta_i} (k_i)) > 2 \alpha / 3$. Suppose that $0 < \varepsilon_i \leq \delta_i$, $\varepsilon_i \to 0$. As the $\varepsilon$-motions appearing in the definition of slow relaxations take the real $(k_i, y_i)$-motions, where $y_i = f(-t_i, y^*, k^*)$, and $t_i$ are determined as follows: $$ t_i = \sup\{ t>0 : \rho (f(t', x, k), f (t' x, k')) < \alpha / 3 \} $$ under the conditions $t' \in [0,t],\ x \in X,\ \rho_K (k, k') < \rho_K (k^*, k_i) \}$. Note that $\rho^*(f(t_i, y_i, k_i), \omega^{\varepsilon_i} (k_i)) \geq \alpha / 3$, consequently, $\eta^{\varepsilon_i}_3 (f(t,y_i, k_i), \alpha / 4) > t_i$ and $t_i \to \infty$ as $i \to \infty$. The last follows from the compactness of $X$ and $K$ (see the proof of Proposition \ref{P4.1}). Thus, $\eta^0_3$-slow relaxations exist and then $\tau^0_3$-slow relaxations exist too. Lemma \ref{L4.8} and Theorem \ref{T4.2} are proved. \end{proof} \section*{Summary} In Sections \ref{S1}-\ref{S4} the basic notions of the theory of transition processes and slow relaxations are stated. Two directions of further development of the theory are possible: introduction of new relaxation times and performing the same studies for them or widening the circle of solved problems and supplementing the obtained existence theorems with analytical results. Among interesting but unsufficiently explored relaxation times let us mention the approximation time $$ \tau(x,k,\varepsilon) = \inf \{ t \geq 0 : d(\omega (x,k), f ([0,t], x,k)) < \varepsilon \} $$ and the averaging time $$ \tau_{v} (x,k,\varepsilon, \varphi) = \inf \Big\{ t \geq 0 : \Big| \frac{1}{t'} \int^{t'}_0 \ \varphi (f(\tau, x,k)) d \tau - \langle \varphi \rangle _{x,k} \Big| < \varepsilon\mbox{ for } t' > t \Big\}, $$ where $\varepsilon>0$, $\varphi$ is a continuous function over the phase space $X$, $$ \langle \varphi \rangle _{x,k} = \lim_{t\to \infty} \frac{1}{t} \int^t_0 \varphi (f(t, x,k)) d \tau $$ (if the limit exists). The approximation time is the time necessary for the motion to visit the $\varepsilon$-neighbourhood of each its $\omega$-limit point. The averaging time depends on continuous function $\varphi$ and shows the time necessary for establishing the average value of $\varphi$ with accuracy $\varepsilon$ along the trajectory. As the most important problem of analytical research, one should consider the problem of studying the asymptotical behaviour under $T \to \infty$ of ``domains of delay", that is, the sets of those pairs $(x,k)$ (the initial condition, parameter) for which $\tau_i (x,k, \varepsilon) > T$ (or $\eta_i (x,k, \varepsilon) > T)$. Such estimations for particular two-dimensional system are given in the work \cite{[67]}. ``Structurally stable systems are not dense". It would not be exaggeration to say that the so titled work by Smale \cite{[39]} opened a new era in the understanding of dynamics. Structurally stable (rough) systems are those whose phase portraits do not change qualitatively under small perturbations (accurate definitions with detailed motivation see in \cite{[5]}). Smale constructed such structurally unstable system that any other system close enough to it is also structurally unstable. This result broke the hopes to classify if not all then ``almost all" dynamical systems. Such hopes were associated with the successes of classification of two-dimensional dynamical systems \cite{[13],[68]} among which structurally stable ones are dense. There are quite a number of attempts to correct the catastrophic situation with structural stability: to invent such natural notion of stability, for which almost all systems would be stable. The weakened definition of structural stability is proposed in the works \cite{[69],[70],[71]}: the system is stable if almost all trajectories change little under small perturbations. This stability is already typical, almost all systems are stable in this sense. The other way to get rid of the ``Smale nightmare" (the existence of domains of structurally unstable systems) is to consider the $\varepsilon$-motions, subsequently considering (or not) the limit $\varepsilon \to 0$. The picture obtained (even in limit $\varepsilon \to 0$) is more stable than the phase portrait (the accurate formulation see above in Section \ref{S4}). It seems to be obvious that one should first study those (more rough) details of dynamics, which do not disappear under small perturbations. The approach based on consideration of limit sets of $\varepsilon$-motions, in the form stated here was proposed in the paper \cite{[22]}. It is necessary to note the conceptual proximity of this approach to the method of quasi-averages in statistical physics \cite{[72]}. By analogy, the stated approach could be called the method of ``quasi-limit" sets. 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