Steady States for a Dynamical System in 2D

contour plot

root list

representation of some roots

z

1

-1.05

z

2

0.1

Consider a hypothetical dynamical system governed by the following equations:

x

t

=

f

1

(x,y)=x-

z

1

2

y

cosy

,

y

t

=

f

2

(x,y)=-y+xsinx+

z

2

,

where

z

1

and

z

2

are bifurcation parameters that vary between

-2

and

+2

and with values set by the user.

The steady states of this system are solutions of the following system of equations:

f

1

(x,y)=x-

z

1

2

y

cosy=0

,

f

2

(x,y)=-y+xsinx+

z

2

=0

.

The above system of two nonlinear equations exhibits multiple solutions that can all be determined using the built-in Mathematica function ContourPlot [1]. In addition to giving a graphical representation of the contours

f

1

(x,y)

=0

and

f

2

(x,y)

=0

and the intersection points (shown in black), this Demonstration provides the numerical values of all roots for

-10≤x≤10

and

-10≤y≤10

.