Andy Sivak

We consider some examples of dynamical systems on compact spaces, which deal with the structure of sets of attraction of trajectories and formulate some problems implied by known results.

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Let f: I → I be a unimodal map (U-map). We say that a finite family A =|J 0, J 1,..., J n-1} of subintervals of the interval I forms a cycle of intervals of period n if the interiors of J i are mutually disjoint and f(J i ) ⊂ J (i + 1)mod n for all i ∈{0, 1, ..., n- 1}. Denote by A n , = A n (f) the set of cycles of intervals of period n of the map...

If a dynamical system describes a real process or phenomenon, then, as a rule, its properties depend on parameters. Any variation of the parameters inevitably results in a certain perturbation of the trajectories of a dynamical system under consideration. It is worth noting that small changes in the parameters may lead to significant changes in the...

Dynamical systems generated by continuous maps of an interval into itself are characterized by the following important property: The data on the relative location of points of a single trajectory on the interval I may contain much information about the dynamical system as a whole. Clearly, this is explained by the fact that the phase space (the int...

Let f: I → I be a continuous map and let B = {β0, β1, ..., βn-1} be its cycle of period n≥1. One can distinguish between two types of stability of the cycle B, namely, between stability under perturbations of the initial data and stability under perturbations of the map. First, we consider the first type of stability.

Dynamical systems are usually understood as one-parameter groups (or semigroups) f t of maps of a space X into itself (this space is either topological or metric). If t belongs to ℝ or ℝ+, then a dynamical system is sometimes called a flow and if t belongs to ℤ or ℤ+, then this dynamical system is called a cascade. These names are connected with th...

As shown in previous chapters, maps of the interval onto itself exhibit fairly diverse dynamical behavior. Therefore, in studying dynamical systems of this sort, it is natural to decompose the entire set of maps into classes exhibiting “similar” dynamical behavior.

The phase space of dynamical systems under consideration, i.e., the interval I, is endowed with Lebesgue measure. It is thus useful to establish some properties of dynamical systems that are typical with respect to this measure, i.e., properties exhibited by trajectories covering sets of full measure.

Symbolic dynamics is a part of the general theory of dynamical systems dealing with cascades generated by shifts in various spaces of sequences $$\Theta = \left( {...{\theta _{ - 2}},{\theta _{ - 1}},{\theta _0},{\theta _1},{\theta _2},...} \right)\quad or\quad \Theta = \left( {{\theta _0},{\theta _1},{\theta _2},...} \right), $$ where θn are lette...

We find a class of weakly unimodal C ∞ maps of an interval with zero topological entropy such that no such map f is Lyapunov stable on the set Per(f) of its periodic points. This disproves a statement published in several books and papers, e.g., by V.