Investigation of dynamical systems using tools of the theory of invariants and projective geometry

The investigation of nonlinear dynamical systems of the type $\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to some ordinary differential equations of the second order in the form $y''+a_1(x,y)y'^3+3a_2(x,y)y'^2+3a_3(x,y)y'+a_4(x,y)=0$ is done. The main backbone of this investigation was provided by the theory of invariants developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century and the projective geometry of E. Cartan. In our work two, in some sense supplementary, systems are considered: the Lorenz system $\dot{x}=\sigma (y-x), \dot{y}=rx-y-zx,\dot{z}=xy-bz$ and the R\"o\ss ler system $\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b+xz-cz.$. The invarinats for the ordinary differential equations, which correspond to the systems mentioned abouve, are evaluated. The connection of values of the invariants with characteristics of dynamical systems is established.Comment: 18 pages, Latex, to appear in J. of Applied Mathematics (ZAMP

Dunajski generalization of the second heavenly equation: dressing method and the hierarchy

Dunajski generalization of the second heavenly equation is studied. A dressing scheme applicable to Dunajski equation is developed, an example of constructing solutions in terms of implicit functions is considered. Dunajski equation hierarchy is described, its Lax-Sato form is presented. Dunajsky equation hierarchy is characterized by conservation of three-dimensional volume form, in which a spectral variable is taken into account.Comment: 13 page

On vector field defined by the hopf map S3 on S2

секция: Аналитическая теория дифференциальных уравнени