By Zeraoulia Elhadj
This ebook is a accomplished number of recognized effects concerning the Lozi map, a piecewise-affine model of the Henon map. Henon map is likely one of the such a lot studied examples in dynamical platforms and it draws loads of recognition from researchers, but it is tough to research analytically. less complicated constitution of the Lozi map makes it better for such research. The booklet isn't just a very good advent to the Lozi map and its generalizations, it additionally summarizes of significant suggestions in dynamical platforms conception reminiscent of hyperbolicity, SRB measures, attractor kinds, and extra.
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Extra info for Lozi Mappings: Theory and Applications
Example text
Generally, these exponents are not continuous functions (in the variable (xr)0≤r<∞) and they need some conditions of existence (in the sense that these exponents are bounded) of the solution under consideration. , ∑ ë Ni = 0. If the system is dissipative, then n ∑ ëN i i=1 < = 0. 17) where k is the maximum integer such that the sum of the k largest exponents is still non-negative, and in this case DKY is an upper bound for the information dimension of the system as shown in [Kaplan & Yorke (1987)].
A whole picture of this procedure can be seen in what follows. 27) Comprehensive Review of Hyperbolicity, Ergodicity, and Chaos 11 Let θ (x, a, b) ¢ [0, 2π] be the angle between s and u at x in the invariant set. In some cases, θ (x, a, b) = 0 if s and u are identical. 28) be the lower bound of the angle of a trajectory xn+1 = f (xn, a, b). 29) where θ ( f i (x0), a, b) is the angle at a point, θm (x0, a, b, N) is the minimum of the angle for (N + 1) points of the trajectory. 30) The numerical procedure given in [Lai, et al.
2. There is a positive Lyapunov exponent at every orbit. 3. They are expansive and so sensitive with respect to initial data. 4. They have zero volume if the flow is C2. 5. There is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the centerunstable Jacobian. A number of concrete Lorenz-like systems were found and it was proved mathematically that they are robustly transitive and not hyperbolic and they are sensitive to initial conditions as shown in [Afraimovich, et al.