1 Nonlinear dynamical systems
2 Energy flow variables and equations
3 Energy flow characteristic factors and vectors
4 Phase volume strain and its time rate of change
5 Zero-energy flow surface, fixed points and stabilities
- Case 1\( \varvec{y} = {\mathbf{0}} \), representing the origin of phase space, at which E is defined as zero.
- Case 2\( \varvec{f} = {\mathbf{0}} \), implying an equilibrium point of the system.
- Case 3\( P = 0,\quad \varvec{y} \ne {\mathbf{0}} \ne \varvec{f}, \) corresponding to a \( \dot{E} = 0 \) surface.
- if \( \left| {\varvec{y} +\varvec{\varepsilon}} \right| < \left| \varvec{y} \right|, \) then \( E (t,\varvec{y} +\varvec{\varepsilon})< E (t,\varvec{y} ), \) so \( \Delta \dot{E} > 0 \) implies a flow towards the zero-energy flow surface, while \( \Delta \dot{E} < 0 \) indicates a flow backwards from the zero-energy flow surface;
- if \( \left| {\varvec{y} +\varvec{\varepsilon}} \right| > \left| \varvec{y} \right|, \) then \( E(t,\varvec{y} +\varvec{\varepsilon}) > E(t,\varvec{y}), \) so \( \Delta \dot{E} < 0 \) implies a flow towards the zero-energy flow surface, while \( \Delta \dot{E} > 0 \) indicates a flow backwards from the zero-energy flow surface;
- if the flows from both sides of the zero-energy flow surface are toward it, this surface is an attracting surface.