By Michael Blank
This booklet provides the examine of ergodic houses of so-called chaotic dynamical platforms. one of many primary issues is the interaction among deterministic and quasi-stochastic habit in chaotic dynamics and among houses of constant dynamical platforms and people in their discrete approximations. utilizing easy examples, the writer describes the most phenomena identified in chaotic dynamical platforms, learning subject matters akin to the operator technique in chaotic dynamics, stochastic balance, and the so-called coupled platforms. The final chapters are dedicated to difficulties of numerical modelling of chaotic dynamics.
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Extra info for Discreteness and Continuity in Problems of Chaotic Dynamics
Example text
21 for an application. 24. 1. Construct an analytical proof of Young’s inequality. 2. Assume a, b, c, d > 0 and prove the following: (a) a2 + b2 ≥ 2ab. (b) a4 + b4 ≥ 2a2 b2 . (c) a2 + b2 + c2 ≥ ab + bc + ca. (d) a4 + b4 + c4 + d4 ≥ 4abcd. (e) (a + b)(b + c)(c + a) ≥ 8abc. 3. Use AM–GM to show that n! < n+1 2 n . Prove also that if n > 1 is a natural number, then [45] (2n − 1)!! , where (2n − 1)!! = (2n − 1) · (2n − 3) · (2n − 5) · · · · · 5 · 3 · 1 (2n)!! = (2n) · (2n − 2) · (2n − 4) · · · · · 4 · 2.
3. If f (x) and g(x) are integrable on [a, b] with f (x) ≤ g(x), then b a b f (x) dx ≤ g(x) dx. a 22 2. Methods from the Calculus Proof. With the notation described above, we form Riemann sums: n n f (xi )∆x ≤ i=1 g(xi )∆x. 1. 1 (Simple Estimate). If f (x) is integrable on [a, b] with m ≤ f (x) ≤ M , then b m(b − a) ≤ f (x) dx ≤ M (b − a). 2 (Modulus Inequality). If f (x) is integrable on [a, b], then b b f (x) dx ≤ a |f (x)| dx. a The second corollary follows from the inequalities −|f (x)| ≤ f (x) ≤ |f (x)| and plays the role of the triangle inequality for integrals.
Underestimating an integral. By the quotient rule for differentiation, d dx 1 x x f (u) du 0 ≥0 as required. It is possible to obtain other inequalities involving integrals through an ad hoc consideration of areas bounded by various plane curves. This simple process, reminiscent of the integral test from calculus, is probably best described in an example. 12. The function f (x) = xp , where −1 < p < 0, is strictly decreasing. 4 it is apparent that n+1 1 n xp dx < n kp < xp dx. 0 k=1 Hence, after carrying out the integrations, (n + 1)p+1 − 1 < p+1 n kp < k=1 np+1 .