Lyapunov exponents of the predator prey model Ljapunovovy ...

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23Figure 2: Convergence of H n,300 for Tent and Logistic maps (n = 1, . . . , 500)3.1 Logistic familyWe already know the Lyapunov exponent of the Logistic mapF 4 (x) = 4x (1 − x) ,where x ∈ [0, 1], L (F 4 (x)) = log 2, see section Preliminaries.This map is the special case of the parametric family, given by equation belowF µ = µx (1 − x) ,where x ∈ [0, 1] and µ ∈ [0, 4] is the parameter.In this section we study the behavior of the Logistic map in dependence on its parameterand then show how do the changes of parameter µ affect the Lyapunov exponent.Firstly we define some terms we will use for describing the behavior of F µ .
24Before we set the precise definition of bifurcation let us explain the intuitive understandingof it. Imagine the situation where the population of rabbits is hunted by foxes.If the population growth is bigger than number of rabbits killed by foxes, the populationwill grow to infinity. If the growth is equal to the number of eaten rabbits, the populationwill be in stagnation. And if foxes kill more rabbits than are born, the population willdie out. This situation is very simplified and in nature would never happen but it is niceexample for explaining the bifurcations. We see that there are three different possibilities.The overbreeding, the stagnation and the dying out. Each of them represents differentdynamical behavior of the system. The value at which the behavior is changing is calledbifurcation boundary. Another examples of bifurcations can be found in [29].Definition 3.15 [15, page 60] Let f µ (x) be a parametrized family of functions. Then there is abifurcation at µ 0 if there exists ϵ > 0 such that whenever µ 1 and µ 2 satisfy µ 0 − ϵ < µ 1 < µ 0and µ 0 < µ 2 < µ 0 + ϵ, then the dynamics of f µ1 (x) are different from the dynamics of f µ2 (x).In other words, the dynamics of the function changes when the parameter value crosses throughthe point µ 0 .Remark 3.16 Let f µ be a map dependent on the real parameter µ. If there are two valuesof the parameter µ 1 , µ 2 (suppose µ 1 < µ 2 ) such that f µ1 and f µ2 are not topologicallyequivalent then there exists a bifurcation boundary at some µ b such that µ 1 ≤ µ b ≤ µ 2 .For parameter between zero and one the map F µ has one attracting fixed point x 1 = 0.When µ = 1 the bifurcation occurs, the fixed point loses its stability and another attractingfixed point is born at x 2 = µ−1µ .When µ passes three there comes the period doubling. That does mean that there isa sequence of bifurcation boundaries b n such that if µ passes b k+1 the periodic pointsof period k turn to be unstable and the new stable cycle of period 2k is born. MitchellFeigenbaum has proved in [10] that for sequence {d k } ∞ k=1 where d k is defined bythe following limit existsd k = b k+1 − b kd kδ = lim .k→∞ d k+1Such defined δ is called the Feigenbaum constant. That does mean that there is a valueof parameter where the period doubling ends. The approximate values of Feigenbaumconstant and the end point of period doubling for Logistic family areδ ≈ 4.669202, b F ≈ 3.569945.To study the behavior of the Logistic family for the parameter between the end of theperiod doubling and four, we should restrict the phase space to its core which is definedby 1 1X c = F 2 , F .2 2
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 and 10: 2List of Listings1 Algorithm for co
Page 11 and 12: 41 IntroductionDynamical systems ar
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: 22It is equivalent to|f n (x) − f
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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