Linearization Procedure and Nonlinear Systems of Differential and Difference Equations

Abstract

Carleman [1] showed that an autonomous finite dimensional nonlinear system of differential equations x=f(x) can be embedded into an infinite linear system of differential equations, where the f. are polynomial in x. Recently, several authors [2–7] have taken up this idea and solved nonlinear differential equations within this approach. Steeb and Wilhelm [4] have written the original system with the help of the Kronecker product ⊗ as: x=Ax+B(x⊗x)+..+Z(x⊗x⊗…⊗x), where A,B,..,Z are matrices with constant coefficients. They calculated the time evolution of x⊗x, x⊗x⊗x and so on and obtained an infinite system with simple block structure, whereas the infinite system obtained from Carleman linearization has no simple block structure and must be rearranged before this can be obtained. The infinite system can easily be expressed with the help of Bose operators. The infinite system and the finite system are not equivalent, because the infinite system admits solutions which are not solutions of the finite system. However, if we restrict ourselves to analytical solutions, then the two approaches are equivalent. When we solve the infinite system by Laplace transformation or an exponential ansatz, we are forced to consider carefully the problem of convergence. When we cut off the infinite system at some finite dimension, we do not, in general, find a good approximation, in particular for large times. However, when the original system admits only asymtotically stable stationary solutions, such an approximation can be carried out. Recently, Steeb [7] applied the approach to nonlinear difference equations. The solution of the logistic equation x(t+1)=ax(t)(1−x(t)) (t=1,2,..) for a=4 can be determined. The ansatz xn(t):=(x(t))n leads to the linear infinite system $${x_n}(t + 1) = {a^n}\sum\limits_{r = 0}^n {{{( - 1)}^r}} \left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right){x_{n + r}}(t)$$.