[Introduction | Background | Results | Conclusions]

Background

The model contains 4 state variables x,y,z,w, and seven parameter values, c₁,c₂,,a,b,,A(mp). x is the position, y is the velocity, and z and w are forcing functions. c₂, a, b, , and A are positive constants and , c₁ are constants that can take any value.

The model equations are given below.

x' = y

y' = - x - (c1 + c2(x - y)2)y + aw + bz

z' = z - w - A(z2 + w2)z

w' = z + w - A(z2 + w2)w

To focus our investigation of this model, 5 parameter values were fixed, at:

parameter	value
c2	3
	0.3
a	3
b	0
A	1

The two remaining parameters are varied to observe the behavior of the system.

The equations for x and y are coupled together. x and y also depend upon the values of z and w. z and w only depend on each other. z and w can be decoupled from the first two equations. Using polar coordinates, the asymptotic solutions for z(t) and w(t) are given by

z(t) = 0,
w(t) = 0 when < or = 0;

z(t) = ,
w(t) = when is > 0.

where is a phase angle that depends on the initial conditions of z and w.