Lyapunov exponents of the predator prey model Ljapunovovy ...

72 PreliminariesIn this section we set the basic definitions, lemmas and theorems, which will be neededin later parts of the thesis. In section Elementary dynamics we recall some basic definitionsand theorems to describe properties of discrete dynamical systems (see [7]). In sectionIntroduction to ergodicity we briefly introduce some of the most important ideas of ergodictheory. For more consult [6].2.1 Elementary dynamicsDefinition 2.1 Let X be a compact metric space and f : X → X be a continuous map. Bya discrete dynamical system is an ordered pair (X, f).The set X is called phase space or state space and the map f is called evolution function. Inthis thesis can be found the notation of systems given just by evolution function. In thatcase as the phase space is understood the strongly invariant set of the evolution function.The strongly invariant set is such set M for which holds f (M) = M.Definition 2.2 A point x is called the fixed point of the map f if f (x) = x and by Fix (f) wedenote the set of all fixed points of the map f. A point x is called the periodic point of period n iff n (x) = x and f k (x) ≠ x for 0 < k < n. By Per n (f) we denote the set of all periodic points ofperiod n for map f.By Per (f) we denote the set of all periodic points of f. By a cycle of period n weunderstand the sequence of points x 1 , . . . , x n ∈ Per n (f) such that f (x k ) = x k+1 fork = 1, . . . , n − 1 and f (x n ) = x 1 .For the next definition suppose that Jacobian matrix of f n at x exists.Definition 2.3 Periodic point x ∈ Per n (f) is called hyperbolic if the Jacobian matrix J f n (x) hasno eigenvalues on the unit circle.Definition 2.4 Let x be a hyperbolic periodic point.(i) x is called sink or attracting periodic point if the absolute values of all the eigenvalues ofJ f n are less than one.(ii) x is called source or repelling periodic point if the absolute values of all the eigenvaluesof J f n are greater than one.(iii) x is called saddle point if some of the absolute values of the eigenvalues of J f n are less thanone and some are greater than one.Example 2.5• Let f : R → R, f (x) = x 3 . Then Fix (f) = {0, 1}. Point x = 0 is attracting fixedpoint and x = 1 is repelling fixed point.