Graph of  potential function as a function of variable system parameter p1

>	xmin:=-5;xmax:=5; p1min:=-1.5;p1max:=1.5;

$\mathrm{xmin}:=-5$

$\mathrm{xmax}:=5$

$\mathrm{p1min}:=-1.5$

$\mathrm{p1max}:=1.5$

>

$\mathrm{fx}=\frac{1}{3}{x}^3-\mathrm{p1}x$

 A ball on the surface represents a system state. Unless  p1 changes, the ball moves on a curve -a  cubic parabola.

Graph of system equilibria

If x is a state variable and   fx (x, p )  is potential function of a system where p is vector of parameters,
then the set of possible resulting equilibria to where system can arrive  is implicitly defined as

>	$\mathrm{dfx}:=\frac{d}{dx}\mathrm{fx}=0$

$\mathrm{dfx}:=x^2-\mathrm{p1}=0$

>	implicitplot(dfx,p1=-10..10,x=-5..5,grid=[100,100]);

 
Here we can see, that there are not appearing not dissapearing new equilibria states for p1>0 . One branch represents stable equilibria  while the other branch represents unstable equilibria. The system state (a ball) can move toward instability (fall down) or retunr to a stable equlibria depending on its intial state.

Differential Equation

(A simple dynamics of a ball on a surface of behavior)

>	$\mathrm{DR}:=\mathrm{a2}\left(\frac{d^2}{{dt}^2}x\left(t\right)\right)+\mathrm{a1}\left(\frac{d}{dt}x\left(t\right)\right)+\mathrm{subs}\left(x=x\left(t\right),\frac{d}{dx}\mathrm{fx}\right)=u\left(t\right)$

>	a2:=2:    a1:=1:    p1:=4:  u(t):=0: DR;

Equilibira:

>	xp:=solve(dfx,x);

$\mathrm{xp}:=$

>	$\mathrm{deqns}:=\left\{\frac{d}{dt}\mathrm{x1}\left(t\right)=\mathrm{x2}\left(t\right),\frac{d}{dt}\mathrm{x2}\left(t\right)=\frac{-\mathrm{a1}\mathrm{x2}\left(t\right)-\mathrm{subs}\left(x=\mathrm{x1}\left(t\right),\frac{d}{dx}\mathrm{fx}\right)}{\mathrm{a2}}\right\}$

Gradient field in state space

>	dfieldplot([deqns[1],deqns[2]],[x1(t),x2(t)], t=-1..1,x1=-3..3, x2=-5..5,title=`Fold`,arrows=medium,color=blue);

---------------- Solving De  ---------------------

Initial conditions

>	x0:=-1.99999999999; dx0:=0;

$\mathrm{x0}:=-1.99999999999$

$\mathrm{dx0}:=0$

>	tmax:=60: xmin:=-5:  xmax:=5: dt:=0.05:

>	DEplot(DR,x(t),t=0..tmax,[[x(0)=x0,D(x)(0)=dx0]],x=xmin..xmax,stepsize=dt,color=black);

Graph of  potential function as a function of variable system parameter p2

>	xmin:=-3;xmax:=3; p1:=1; p2min:=-3;p2max:=1;

$\mathrm{xmin}:=-3$

$\mathrm{xmax}:=3$

$\mathrm{p1}:=1$

$\mathrm{p2min}:=-3$

$\mathrm{p2max}:=1$

>	'fx'=fx; plot3d(fx,x=xmin..xmax,p2=p2min..p2max,grid=[100,100]);

$\mathrm{fx}=\frac{1}{4}{x}^4+x+\frac{1}{2}\mathrm{p2}{x}^2$

 On this surface, we may imagine a ball whose position represents a state of a system. Unless p2 changes, a ball moves on a curve given by 4-order polynomial.

Graph of system equilibria

 If x is a state variable and   fx (x, p )  is potential function of a system where p is vector of parameters,
then the set of possible resulting equilibria to where system can arrive  is implicitly defined as

>	$\mathrm{dfx}:=\frac{d}{dx}\mathrm{fx}=0$

$\mathrm{dfx}:=x^3+\mathrm{p1}+\mathrm{p2}x=0$

>	implicitplot3d(dfx,p1=-1..1,p2=-3..1,x=-3..3,grid=[30,30,30]);

 Plocha určuje možné rovnovážné stavy systému (pozor, stabilní i nestabilní).

 
Fig .2: a 3: Adopted from Catastrophe theory by E.C. Zeeman, Trinity University, San Antonio, March 1995,http://zakuski.utsa.edu/~gokhman/ecz/c.html

Differential Equation

A simple dynamics of a ball on a surface of behavior

>	$\mathrm{DR}:=\mathrm{a2}\left(\frac{d^2}{{dt}^2}x\left(t\right)\right)+\mathrm{a1}\left(\frac{d}{dt}x\left(t\right)\right)+\mathrm{subs}\left(x=x\left(t\right),\frac{d}{dx}\mathrm{fx}\right)=u\left(t\right)$

$\mathrm{DR}:=\mathrm{a2}\left(\frac{d^2}{{dt}^2}x\left(t\right)\right)+\mathrm{a1}\left(\frac{d}{dt}x\left(t\right)\right)+{x\left(t\right)}^3+\mathrm{p1}+\mathrm{p2}x\left(t\right)=u\left(t\right)$

>	a2:=2:    a1:=1:    p1:=0.5:  p2:=-3: u(t):=0: DR;

$2\left(\frac{d^2}{{dt}^2}x\left(t\right)\right)+\frac{d}{dt}x\left(t\right)+{x\left(t\right)}^3+.5-3x\left(t\right)=0$

Equilibira::

>	xp:=solve(dfx,x);

$\mathrm{xp}:=-1.810037929,.1682544018,1.641783528$

>	$\mathrm{deqns}:=\left\{\frac{d}{dt}\mathrm{x1}\left(t\right)=\mathrm{x2}\left(t\right),\frac{d}{dt}\mathrm{x2}\left(t\right)=\frac{-\mathrm{a1}\mathrm{x2}\left(t\right)-\mathrm{subs}\left(x=\mathrm{x1}\left(t\right),\frac{d}{dx}\mathrm{fx}\right)}{\mathrm{a2}}\right\}$

$\mathrm{deqns}:=\left\{\frac{d}{dt}\mathrm{x1}\left(t\right)=\mathrm{x2}\left(t\right),\frac{d}{dt}\mathrm{x2}\left(t\right)=-\frac{1}{2}\mathrm{x2}\left(t\right)-\frac{1}{2}{\mathrm{x1}\left(t\right)}^3-.2500000000+\frac{3}{2}\mathrm{x1}\left(t\right)\right\}$

Gradient field in state space

>	dfieldplot([deqns[1],deqns[2]],[x1(t),x2(t)], t=-2..2,x1=-2.5..2.5, x2=-2..2,title=`Cusp`,arrows=medium,color=blue);

Do you see the equilibrium points in gradient field?

----------------  Solution of DE  ---------------------

Ininital Conditions

>	x0:=-3; dx0:=0;

$\mathrm{x0}:=-3$

$\mathrm{dx0}:=0$

>	tmax:=60: xmin:=-5:  xmax:=5: dt:=0.05:

>	DEplot(DR,x(t),t=0..tmax,[[x(0)=x0,D(x)(0)=dx0]],x=xmin..xmax,stepsize=dt,color=black);