Lyapunov exponents of the predator prey model Ljapunovovy ...

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39Figure 17: Bifurcation diagram of Lotka–Volterra family for µ = 0.75Figure 18: Bifurcation diagram of Lotka–Volterra family for µ = 1.0Lyapunov exponents for parameters µ ∈ [0, 1] and ξ ∈ [0, 1] are in Figures 20 and 21.Let us note that for ξ = 0 the maximal Lyapunov exponents of S µ,ξ correspond to theLyapunov exponents of Logistic family F µ to which the restriction of S µ,ξ is conjugate.In Figure 22 is plotted the sign of the maximal Lyapunov exponent L 1 . Blue colorrepresents negative Lyapunov exponents and red color represents the positive ones. Hereit is easy to see for which choice of parameters the system is showing chaotic motion, and
40(a) µ = 0.25 (b) µ = 0.5(c) µ = 0.75 (d) µ = 1.0Figure 19: Lyapunov exponents of Lotka–Volterra family for fixed a.for which one is the system exhibiting periodic or quasi-periodic movement. E.g. thechaotic situation is on Figure 19d for parameter values µ = 1, ξ = 1 and the other is onthe Figure 19b for values of parameters µ = 0.5, ξ = 0.2.Notice the parametrized model, that is researched, has identical dynamical propertieslike F 4 (x) for ξ = 0. That clearly follows from the fact that the second iteration equals toF 4 (x), namelyS µ,0 (x, y) = (µx (4 − x) − xy, 0)andS 2 µ,0 (x, y) = (µα (4 − α) − α · 0, 0) = (µα (4 − α) , 0)
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 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: 38Open problem 5.1 As simulations s
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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