In order to investigate the geometrical relation between two flows in two dynamical systems, a flow for an investigated dynamical system is called the compared flow and a flow for a given dynamical system is called the reference flow. A surface on which the reference flow lies is termed the reference surface. The time-change rate of the normal distance between the reference and compared flows in the normal direction of the reference surface is measured by a new function (i.e., $G$ function). Based on the surface of the reference flow, the $kth$-order $G$ functions are introduced for the noncontact and $lth$-order contact flows in two different dynamical systems. Through the new functions, the geometric relations between two flows in two dynamical systems are investigated without contact between the reference and compared flows. The dynamics for the compared flow with a contact to the reference surface is briefly addressed. Finally, the brief discussion of applications is given.

1.
Kreyszig
,
E.
, 1959,
Differential Geometry
,
University of Toronto Press
,
Toronto
.
2.
Luo
,
A. C. J.
, 2005, “
A Theory for Nonsmooth Dynamic Systems on the Connectable Domains
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
10
, pp.
1
55
.
3.
Luo
,
A. C. J.
, 2006,
Singularity and Dynamics on Discontinuous Vector Fields
,
Elsevier
,
Amsterdam
.
4.
Luo
,
A. C. J.
, 2007, “
A Theory for n-Dimensional Nonlinear Dynamics on Continuous Vector Fields
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
11
, pp.
117
194
.
5.
Luo
,
A. C. J.
, 2007,
Global Transversality, Resonance and Chaotic Dynamics
,
World Scientific
,
Singapore
.