C2 FUNDAMENTAL THEORY OF DYNAMICAL SYSTEMS
C3 THEORY OF COMPUTATIONAL DYNAMICS
HANDOUT 4
LINEARIZING ABOUT TRAJECTORIES
Let ϕt denote the solution of the differential equation:
dx(t )
dt
=
F(x(t))
(1)
dϕ t (x )
dt
=
F(ϕt(x))
(2)
so that
Let x t = ϕt(x) be a trajectory with initial condition x at t = 0 and vt be a small perturbation to xt with
initial condition v = v0, so that
ϕt(x + v)
=
ϕt(x) + vt
Thus
vt
=
=
ϕt(x + v) - ϕt(x)
2
ϕt(x) + Dxϕt.v + O( v ) - ϕt(x)
=
D xϕt.v + O( v
2
)
Thus to first order, D xϕt maps initial perturbations to final perturbations. To see how to compute
D xϕt, we first derive a differential equation for vt. We have
dv t
dt
=
d
d
ϕt(x + v) ϕt(x)
dt
dt
=
F(ϕt(x + v)) - F(ϕt(x))
=
=
F(ϕt(x) + vt) - F(ϕt(x))
F(ϕt(x)) + D x F.vt + O( vt
=
t
D x F.vt + O( vt
t
2
) - F(ϕt(x))
2
)
Thus, to first order, vt satisfies the differential equation
dv t
dt
=
D x F.vt
t
(3)
This is known as the linearization of (1). Now denote M t = D xϕ t, so that v t = M t.v. Substituting this
into (3), we have
dM t .v
dt
=
D x F.Mt.v
t
This holds for all initial v, and hence
Linearizing Flows
dM t
dt
=
2
D x F.Mt
(4)
t
We thus see that the matrix Mt satisfies the same differential equation as each individual perturbation vt. To see why this should be so, let e(1), … , e(n) be the standard basis of unit vectors in Rn. Each
e (i) evolves according to (3), that is
de t(i )
dt
=
(i)
D x F.e t
t
(1)
(n)
Thus the matrix whose columns are e t , …, e t must also satisfy this equation. But by the definition
of Mt
(i)
et
=
Mt.e (i)
(1)
(n)
and for any matrix B the ith column is just B.e (i). Thus the matrix whose columns are e t , …, e t is
precisely Mt itself, and hence Mt satisfies (3).
An alternative way of deriving (4) which is much slicker, but perhaps less informative is simply to
differentiate (2) with respect to x:
Dx
dϕ t (x )
dt
=
D x(F(ϕt(x)))
Using the chain rule we get:
Dx
dϕ t (x )
dt
=
D x F.D xϕt,
t
Assuming that ϕt is sufficiently regular to allow interchange of the order of differentiation:
d
D xϕt
dt
as required.
=
D x F.D xϕt,
t