Lyapunov exponents of the predator prey model Ljapunovovy ...

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AbstraktCílem této práce je indikovat a kvantifikovat chaos v diskrétních dynamických systémechzávislých na reálných parametrech. Nejprve je studován Ljapunovův exponent jednorozměrnýchzobrazení. Nejprve je podrobně zkoumán Logistický parametrický systém,jehož Ljapunovovy exponenty jsou počítány. Následně jsou uvedeny a studovány Ljapunovovyexponenty pro n-rozměrná zobrazení. Konkrétně je studováno Hénonovo zobrazení.Cílem zkoumání je vyšetření parametrického systému odvozeného od Lotka–Volterra zobrazení. K výpočtům a simulacím Ljapunovových exponentů byly užity algoritmy,uvedené v závěru práce.Klíčová slova: Ljapunovův exponent, chaos, ergodicita, Logistické zobrazení, Hénonovozobrazení, Lotka–VolterraAbstractThe main aim of this thesis is to indicate and quantify chaos in discrete dynamical systemsthat are depending on real parameters. As a first step the Lyapunov exponentis studied for one-dimensional maps. More precisely, the Logistic parametric family isdeeply researched and the Lyapunov exponents are computed. Secondly, the Lyapunovexponents are introduced, studied and computed for n-dimensional maps, the Hénonmap is investigated. As a goal of the research the Lotka–Volterra parametric family is introducedand utilizing installed algorithms their Lyapunov exponents are simulated andevaluated.Keywords:VolterraLyapunov exponent, chaos, ergodicity, Logistic map, Hénon map, Lotka–
List of Used Abbreviations and SymbolsC n (X) – set of all n times continuously differentiable maps on Xf i – i-th iteration of the map ff| S– restriction of f to SR – set of all real numbersR n – set of all n-dimensional real vectors
Page 1: VSB - Technical University of Ostra
Page 5: I would like to express my thanks t
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 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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