Geometrical properties of the multidimensional nonlinear differential equations and the Finsler metrics of phase spaces of dynamical systems
The first part of the paper is devoted to describing some nonlinear differential equations arising as conditions on the curvature tensors of 3- and 4-dimensional Riemannian manifolds. The metrics \(ds\) are assumed to be of the standard forms \(ds^2 = \Phi^2dt^2 + A^2dx^2 + B^2dy^2 + C^2dz^2\), etc. In the second part some questions of the Cartan theory for equations \(y''=f(x,y,y')\) and geometrical properties of the space of linear elements \((x,y,y')\) are discussed and examples of Finsler metrics for the second order differential equation \(y''=3(y')^2/y + (1/x-my)y'+ (nx^3-1/x)y^4 + (nx^2+t) y^3-sy^2/x\) are constructed. This equation is equivalent to E. Lorentz's nonlinear dynamical system \(\dot x=k(x-y)\), \(\dot y=rx-y-zx\), \(\dot z=xy-bz\). Those examples are found from special Finsler metrics \(ds= [a(x,y)dy + b(x,y) dx]^r dx^s\), \(r+s=1\), which give rise to Berwald spaces with the main scalar \(I\) satisfying \(I^2>4\).
scientific article; zbMATH DE number 907304