Generalized System of Riccati-Type Equations

A new system of generalized Riccati-type equations is derived. An interconnection between the solutions of n-th order differential equations and the solutions of a generalized system of Riccati-type equations is established. Inverse mapping from the solutions of generalized Riccati-type equations onto the linearly independent solutions of the n-th order differential equation is constructed.


Introduction
The structure of the standard Riccati equation is defined in terms of a first order derivative and a second order polynomial. The Riccati equation is associated foremost with differential equation and the Möbius transformation [1]. Analogously, the generalized system of n-th order Riccati-type equations is also associated with the n-th order differential equations. In the present paper this problem is formulated as follows.
Consider the evolution equation with respect to a parameter t generated by the finite dimensional operator H d dt the direct closed-form solution of which is given by the formula The finite dimensional operator H is represented by an n × n matrix which obeys the characteristic polynomial equation As a matter of convenience let us suppose that the characteristic polynomial coincides with the minimal polynomial Let E be a companion matrix of the polynomial f (X) associated with operator H. The companion matrix obeys the same characteristic equation, namely, e-mail: yamaleev@jinr.ru Besides the evolution equation generated by the operator H one may consider an evolution generated by the polynomial f (X). This evolution is described by the n-th order Riccati equation The aim of the present contribution is to establish a mapping between solutions of the equations (1) and (5).

Generalized trigonometric functions as solutions of high-order Riccati equation
In the same way as the usual complex algebra induces the trigonometry, the general complex algebra GC n induces representations of the set of generalized trigonometric functions [2], [3]. A matrix representation of the GC n algebra is given by the companion matrix. The companion matrix E is the representation of the equivalent class of all n × n matrices with trace a 1 , determinant a n and the sum of corresponding minors a i , 2, . . . , n − 1. The explicit form of this matrix is defined as follows An analogy of the Euler formula is defined by the series where the polynomial Q(U) denotes an (n − 1)-degree polynomial of the form The parameter φ stands for the set of (n − 1) parameters φ := (φ 1 , φ 2 , φ 3 , . . . , φ n−1 ). The structure of the set of differential equations for generalized trigonometric functions g 0 (φ), g 1 (φ), g 2 (φ), . . . , g n−1 (φ) is governed by the matrix E and its degrees E k , k = 1, . . . , n − 1 formulated in the standard way: where v g (φ) means a vector of components v g = [g 0 , g 1 , g 2 , . . . , g n−1 ] T .
As proved in [4], the differential equations (2.9) are reduced to an n-th order Riccati equation In this approach the solution of the n-th order Riccati equation is defined as a fraction of two trigonometric functions where φ n−1 depends of (n − 2) parameters φ n−1 (φ 1 , φ 2 , . . . , φ n−2 ), this dependence in a implicit way is defined by the constraints (12). The transformation of the linear system of evolution equations into the canonical form of the n-th order Riccati equation requires the n − 2 constraints (12). Under these constraints the polynomial Q(U) of order (n − 1) is reduced to a linear function of the form Then, the solution of the equation Q(U) = 0 turns out to the solution to the n-th order Riccati equation (11). This observation prompts us the idea to seek differential equations for the roots of the polynomial Q(U) of order (n − 1). As a result, we get a system of Riccati-type equations for the functions where u k are roots of the polynomial Q(u k ).
The explicit form of the system of equations (19) is presented as follows If the basic polynomial f (X) is a polynomial of the n-th order then the functions u k , k = 1, . . . , n − 1 are roots of the following polynomial of the order (n − 1): Each of the roots u k obeys the system of differential equations (20), where and

Conclusion
We have derived the system of Riccati-type equations from the linear system of evolution equations. The method can be applied in the problem of transformation of the n-th order differential equation into a Riccati-type equation. It is expected, the present method will be useful in the theory of finite dimensional quantum mechanics.

Acknowledgments
The author is greatly indebted to Professor Giuseppe Dattoli for his attentive reading of this work and generous encouragements. Author thanks Dr. Adan Rodriguez-Dominguez for fruitful discussions.