Lyapunov exponents of the predator prey model Ljapunovovy ...

173 Lyapunov exponents for one-dimensional mapsIn this section Lyapunov exponents are computed for one-dimensional maps. We introducea numerical approximation of Lyapunov exponent and compute Lyapunov exponentsof well-known maps such as Tent map or Logistic map. Here we show also whatthe Lyapunov exponent describes and how it depends on the behavior of the dynamicalsystem.Definition 3.1 Let X = [a, b] , a, b ∈ R be an interval, µ is an absolutely continuous invariantmeasure on that interval and f is a piecewise continuously differentiable map on X. The numberis called the Lyapunov exponent of f.L (f) = balog f ′ dµThe alternative definition of Lyapunov exponent isn−11 L (f) = lim log f ′ f k (x)a.e.n→∞ nk=0which can be obtained from Definition 3.1 by using the Birkhoff Ergodic Theorem (Theorem2.22). This definition is often used since computing the integral in the Definition 3.1can be hard.The Lyapunov exponent is often used for indication of sensitive dependence on initialconditions. The system with negative Lyapunov exponent is considered stable and thesystem with positive Lyapunov exponent is considered unstable or chaotic. This isn’t truefor general maps. In [18] was proved that positive Lyapunov exponent implies sensitivedependence on initial conditions under some additional conditions for continuous mapson the closed interval (see Proposition 3.3). Moreover in [18] it was shown that the map isLyapunov stable if the Lyapunov exponent is negative taking into account continuouslydifferentiable maps on the closed interval (see Proposition 3.2).Proposition 3.2 Suppose f : [0, 1] → [0, 1] is C 2 . If the Lyapunov exponent L (f) < 0 forsome x 0 , then the orbit of x 0 is Lyapunov stable.Proposition 3.3 Suppose f : [0, 1] → [0, 1] is C 2 . If the Lyapunov exponent L (f) > 0 forsome x 0 and the orbit of x 0 satisfiesinf f ′ (x n ) > 0,n≥0then the orbit exhibits sensitive dependence on initial conditions.Open question 3.4 (i) If a continuous map on interval has positive Lyapunov exponent,does it imply chaos? If it does, in which sense?