Axiomatic theory of scalar fields dynamic systems

In the article the collective name «theory just as the theory of scalar elds of dynamical systems or the problem of adaptive control by initial state replaced» by a more informative (in the title of the article). New concepts of dynamical systems theory (scalar field, the center offset of the scalar field) to a single unified space of dimension (3n+2), where 1) n+1 - dimensionality of the state space; 2) n is the dimensionality of the decision space (phase space); 3) (n+1)- dimension of hyperspace-state center displacement (second and third components of a single space called the space of possible States. In a hypothetical example, solved the problem of adaptive control points pulled out of hyperspace. Proved the Pythagorean theorem for everywhere undifferentiated trigonometric functions K. Weierstrass. In the language of classical proportions and sequences of the «Golden section» three new proven formula that helps to identify dynamical chaos as the automorphism (or self-similarity) analytic geometry. The automorphism demonstrated new algorithms for the synthesis of sequences of Lucas and D. Bernoulli (Binet). New mathematical formula synthesized in the language of calculus mathematical operations by means of the exponent (the formalism of tuple and mask), that allows to speak about a new scientific direction, named to the calculation of mathematical formulas. In conclusion, the article made the world's first animation of the motion of the center of displacement of the attractor of Rossler, describing chemical and physical processes of mixing in a mixture, and the specified Internet address Yandex-cloud, on which this animation is available to everyone.