constraints = list or set of equations and inequalities

evaluationlimit = posint

functions = list of lists

initialpoint = list or set of equations

If only some of the states and inputs are specified by the initialpoint, the EquilibriumPoint command sets the rest of the states and inputs equal to zero.

If the initialpoint is not given, the EquilibriumPoint command chooses the initial point randomly.

simplify = truefalse

The EquilibriumPoint command finds the equilibrium point of eqs such that constraints, if specified by the constraints option, are satisfied, and returns four lists of equations specifying the values of states, derivatives of states, inputs, and outputs at the equilibrium point, respectively.

The equilibrium point of a system is a point at which derivatives of the states vanish. The EquilibriumPoint command performs a local search and returns an equilibrium point closest to the initial point, either specified by the initialpoint parameter or chosen randomly.

If the EquilibriumPoint command cannot find a point at which derivatives are zero, it returns a point that minimizes the derivatives. It is possible to prescribe a non-zero value to the derivatives using the optional parameter constraints.

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\:$

$\mathrm{sys1}≔\left[\mathrm{diff}\left(x\left[1\right]\left(t\right)\,t\right)={x\left[2\right]\left(t\right)}^2-4\,\mathrm{diff}\left(x\left[2\right]\left(t\right)\,t\right)=x\left[1\right]\left(t\right)-1+u\left(t\right)\,y\left(t\right)=x\left[1\right]\left(t\right)+x\left[2\right]\left(t\right)\right]$

$\mathrm{sys1}≔\left[\frac{\ⅆ}{\ⅆt}{x}_1\left(t\right)={x_2\left(t\right)}^2-4\,\frac{\ⅆ}{\ⅆt}{x}_2\left(t\right)=x_1\left(t\right)-1+u\left(t\right)\,y\left(t\right)=x_1\left(t\right)+x_2\left(t\right)\right]$

$\mathrm{EquilibriumPoint}\left(\mathrm{sys1}\&comma;\left[u\left(t\right)\right]\&comma;\mathrm{constraints}=\left[0<x\left[1\right]\left(t\right)\right]\&comma;\mathrm{initialpoint}=\left[u\left(t\right)=0\&comma;x\left[1\right]\left(t\right)=2\&comma;x\left[2\right]\left(t\right)=4\right]\right)$

$\left[x_1\left(t\right)=1.49999999768708\&comma;x_2\left(t\right)=\mathrm{-2.00000001053627}\right],\left[\frac{\&DifferentialD;}{\&DifferentialD;t}{x}_1\left(t\right)=4.21450963017378\times {10}^{\mathrm{-8}}\&comma;\frac{\&DifferentialD;}{\&DifferentialD;t}{x}_2\left(t\right)=\mathrm{-4.62584648364128}\times {10}^{\mathrm{-9}}\right],\left[u\left(t\right)=\mathrm{-0.500000002312923}\right],\left[y\left(t\right)=\mathrm{-0.500000012849197}\right]$

sys2 := {piecewise(x[1](t)<0, x[1](t), x[2](t) + x[1](t)^2) * piecewise(u(t)<0, cos(y(t)), sin(y(t))) = sin(x[1](t)^2) + 5 * y(t) + diff(x[1](t), t, t), y(t) - x[1](t)^2 + u(t)*x[1](t), diff(x[2](t), t) = f(x[1](t), u(t))};
user_function := [
    f,
    [float, float],
    float,
    proc(x, y)
    local d1, d2;
        d1 := cos(x)+x^2;
        d2 := y*d1 + y^2;
        return d1*x+d2*y- exp(d1);
    end proc
    ];

$\mathrm{sys2}≔\left\{y\left(t\right)-{x_1\left(t\right)}^2+u\left(t\right)x_1\left(t\right)\&comma;\left(\left\{\begin{array}{cc}{x}_1\left(t\right)& x_1\left(t\right)<0\\ x_2\left(t\right)+{x_1\left(t\right)}^2& \mathrm{otherwise}\end{array}\right.\right)\left(\left\{\begin{array}{cc}\cos \left(y\left(t\right)\right)& u\left(t\right)<0\\ \sin \left(y\left(t\right)\right)& \mathrm{otherwise}\end{array}\right.\right)=\sin \left({x_1\left(t\right)}^2\right)+5y\left(t\right)+\frac{{\&DifferentialD;}^2}{\&DifferentialD;t^2}{x}_1\left(t\right)\&comma;\frac{\&DifferentialD;}{\&DifferentialD;t}{x}_2\left(t\right)=f\left(x_1\left(t\right)\&comma;u\left(t\right)\right)\right\}$

$\mathrm{user\_function}≔\left[f\&comma;\left[\mathrm{float}\&comma;\mathrm{float}\right]\&comma;\mathrm{float}\&comma;\mathbf{proc}\left(x\&comma;y\right)\mathbf{local}\mathrm{d1}\&comma;\mathrm{d2}\&semi;\mathrm{d1}≔\cos \left(x\right)\&plus;x\&Hat;2\&semi;\mathrm{d2}≔y\&ast;\mathrm{d1}\&plus;y\&Hat;2\&semi;\mathbf{return}x\&ast;\mathrm{d1}\&plus;\mathrm{d2}\&ast;y-\exp \left(\mathrm{d1}\right)\mathbf{end\; proc}\right]$

$\mathrm{EquilibriumPoint}\left(\mathrm{sys2}\&comma;\left[u\left(t\right)\right]\&comma;\mathrm{functions}=\left[\mathrm{user\_function}\right]\&comma;\mathrm{initialpoint}=\left[x\left[1\right]\left(t\right)=1\&comma;x\left[2\right]\left(t\right)=1\&comma;u\left(t\right)=1\right]\right)$

$\left[x_1\left(t\right)=0.853420831346895\&comma;x_2\left(t\right)=0.687832129312234\right],\left[\frac{{\&DifferentialD;}^2}{\&DifferentialD;t^2}{x}_1\left(t\right)=2.43748043970982\times {10}^{\mathrm{-9}}\&comma;\frac{\&DifferentialD;}{\&DifferentialD;t}{x}_1\left(t\right)=0.\&comma;\frac{\&DifferentialD;}{\&DifferentialD;t}{x}_2\left(t\right)=\mathrm{-1.14152598484907}\times {10}^{\mathrm{-9}}\right],\left[u\left(t\right)=1.07055911392119\right],\left[y\left(t\right)=\mathrm{-0.185310333631790}\right]$

The evaluationlimit option was introduced in Maple 15.

For more information on Maple 15 changes, see Updates in Maple 15.