274 CHAPTER 12. SPECTRAL THEORY
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Another very different kind of behavior is also observed. It is possible for the orderedpairs to spiral around the origin.
Example 12.2.9 Suppose a dynamical system is of the form(x(n+1)y(n+1)
)=
(0.7 0.7−0.7 0.7
)(x(n)y(n)
)Find solutions to the dynamical system for given initial conditions.
In this case, the eigenvalues are complex, .7 + .7i and .7− .7i. Suppose the initialcondition is (
x0
y0
)what is a formula for the solutions to the dynamical system? Some computations show thatthe eigen pairs are (
1i
)←→ .7+ .7i,
(1−i
)←→ .7− .7i
Thus the matrix is of the form(1 1i −i
)(.7+ .7i 0
0 .7− .7i
)(12 − 1
2 i12
12 i
)and so,(
x(n)y(n)
)=
(1 1i −i
)((.7+ .7i)n 0
0 (.7− .7i)n
)(12 − 1
2 i12
12 i
)(x0
y0
)The explicit solution is given by
x = x0
(12((0.7−0.7i))n +
12((0.7+0.7i))n
)+y0
(12
i((0.7−0.7i))n− 12
i((0.7+0.7i))n)
y = y0
(12((0.7−0.7i))n +
12((0.7+0.7i))n
)−x0
(12
i((0.7−0.7i))n− 12
i((0.7+0.7i))n)