Lyapunov exponents of the predator prey model Ljapunovovy ...

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3ContentsContents 31 Introduction 42 Preliminaries 72.1 Elementary dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Introduction to ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Lyapunov exponents for one-dimensional maps 173.1 Logistic family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Lyapunov exponents for n-dimensional maps 274.1 Hénon map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Lyapunov exponents of a Lotka–Volterra systems 366 Implementation 437 Conclusions 52References 53
41 IntroductionDynamical systems are often used to describe phenomena in various fields of sciencelike physics, chemistry, biology, economy etc. Many years scientists believed that behaviorof systems given by differential or difference equations can be totally described asperiodic or quasi-periodic. They believed that systems, which behavior is unpredictable,are just very complex. Later it was shown that such unpredictability can occur in simplesystems and new phenomenon, chaos, was described. In this section we briefly show themost important events in the history of chaos theory.From 1885 H. Poincaré studied the three-body problem. This problem describes themotion of bodies in simplified solar system [37]. The first and crucial development inthe field of dynamical systems was made by H. Poincaré in 1890 [26] by recurrence, thatis a point returns to itself arbitrarily close under the actions (iterations), or equivalentlythe point belongs to its omega limit set. Hence, a dynamical system preserves volume,all trajectories return arbitrarily close to their initial position and they do this an infinitenumber of times. More precisely, H. Poincaré discovered: If a flow preserves volume andhas only bounded orbits then for each open set there are orbits that intersect the set infinitely often.The Lyapunov stability, named after Aleksandr Lyapunov, describes the behavior ofpoints in the neighborhood of the equilibria [23]. The equilibrium is Lyapunov stable iforbits of all the points in the neighborhood of that equilibrium don’t leave the neighborhood.The equilibrium is asymptotically stable if orbits of all points that are in the neighborhoodof that equilibrium converge to the equilibrium. Stronger notion is exponentialstability which guarantees the minimal convergence speed. Later Lyapunov extendedhis studium of neighboring points to any two infinitesimally close points. The notion ofLyapunov exponent was set to measure the rate at which orbits of two infinitesimallyclose points diverge.In the first half of the next century the ergodic theory studied dynamical systems motivatedby some physical and mechanical problems. In 1920 G. D. Birkhoff provide thetheorem which says that in dynamical systems given by continuous and transitive mapthere exists a set that has a dense orbit in the phase space. The notion of transitivity wasintroduced by G. D. Birkhoff in 1920 for flows [5]. The map F is transitive if for any twonon-empty open sets U, V , that are subsets of the phase space, there is n ∈ N such thatF n (U) ∩ V ≠ ∅.But the most important idea in ergodic theory came in 1931. It was the Birkhoff ErgodicTheorem [6, Chapter 3] which says that for given integrable function f and measurepreserving transformation T is the average of f along the orbit of T equal to the integralof f.The foundation of the chaos theory, as it is known today, was in 1975 when T. Y. Li andJ. A. Yorke set the definition of chaos [22]. The map is chaotic in sense of Li and Yorke if
Page 1: VSB - Technical University of Ostra
Page 5 and 6: I would like to express my thanks t
Page 7 and 8: List of Used Abbreviations and Symb
Page 9: 2List of Listings1 Algorithm for co
Page 13 and 14: 6In the section Lyapunov exponents
Page 15 and 16: 8• Let f : R 2 → R 2 , f (x, y)
Page 17 and 18: 10Let E = [0.4, 0.6]. Thenλ (E) =
Page 19 and 20: 12for n ≥ 1 and k ∈ Z. Then {E
Page 21 and 22: 14Proof. Let g = h − α. Then B
Page 23 and 24: 16Summing over k we obtain g ∗ d
Page 25 and 26: 18(ii) If a continuous map on inter
Page 27 and 28: 20Substituting t = θ/2, we obtain
Page 29 and 30: 22It is equivalent to|f n (x) − f
Page 31 and 32: 24Before we set the precise definit
Page 33 and 34: 26Figure 4: Bifurcation diagram (bl
Page 35 and 36: 28Definition 4.2 For x ∈ X and v
Page 37 and 38: 30Figure 6: Convergence of log σ n
Page 39 and 40: Figure 10: Bifurcation diagram of H
Page 41 and 42: 34Figure 12: Lyapunov exponent L 1
Page 43 and 44: 365 Lyapunov exponents of a Lotka-V
Page 45 and 46: 38Open problem 5.1 As simulations s
Page 47 and 48: 40(a) µ = 0.25 (b) µ = 0.5(c) µ
Page 49 and 50: 42Figure 22: Sign of the maximal Ly
Page 51 and 52: 441920 # create the orbit of seeds2
Page 53 and 54: 463031 # compute the product of Jac
Page 55 and 56: 18 x = [[ Decimal(0.0),Decimal(0.0)
Page 57 and 58: 34 # create the orbits35 for s in r
Page 59 and 60: 527 ConclusionsThe main aim of this
Page 61 and 62:
54[17] T. Kapitaniak, Chaos for eng
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