Tutorial 9.5: Nonlinear systems
Consider the system of ODEs:
ẋ = f (x, y) = x(3 − x − 2y)
ẏ = g(x, y) = y(2 − x − y)
Finding steady-states The steady states (x∗ , y ∗ ) satisfy f (x∗ , y ∗ ) = g(x∗ , y ∗ ) = 0.
1. Find the four steady-states of the system ODE above.
The linearized system is
d
dt
u
=
v
∂f
∗ ∗
∂x (x , y )
∂g
∗ ∗
∂x (x , y )
∂f
∗ ∗
∂y (x , y )
∂g
∗ ∗
∂y (x , y )
! u
u
.
=A
v
v
A is called the Jacobian matrix evaluated at the steady-state (x∗ , y ∗ ) of the system. The linearized system describes the behaviour of the system near each of the steady-states.
2. Find the Jacobian matrix for the system above.
3. Evaluate the Jacobian matrix at each of the steady-states found above.
1
, Ӽ 4uӼӼӼ‚ ªӼ4!ǟ//ӼӼ \Є ӼӼ ! 4Ӽ, Ӽ Ӽ+ӼӼe
++ Ӽ Ӽ 3 ƒӼ
FӼ + ӼӼ+‚ Ӽ Ӽ+ӼӼ 4 -ZӼ,Ӽ /)Ӽ ˅˅ z͙ Ӽ ӢӼ˅ z͙ -!/ ƥ
! /4ƒӼ !Ӽ +‚ Ӽ Ӽ Ӽ ΒӼ 9< ĹŨ˅ UWZ b[ Z ӼUlZ W[ZӼ Ӽ U³Z ³µSӼ FӼ / +4Ӽ
-ªӼ,Ӽ-!ӼӼ¾ ǶӼ
r
ğ ə ɗɘ
describes
Sketching the phase portrait The linearized system
the behaviour
ɗɘ
əӼ
ƋbӼ of the system
´ ¶̵͙ ¶̾ ´͙ l ¸ b Ƌ bîӼ
& & ¡£ ¢£ ¸
ƒӼ
near each of the steady-states. For example, for the steady-state (0,¦îӼ
0),
b ¦ ϕ˚¦ bîӼ
3 0ӼӼ+!Ӽ+‚ Ӽ ӼӼ!ѭΓӼ
{,Ӽ
A=
,
0 2
9< V ȅ˅ FӼ &͙ ˅
brcXr
so the eigenvalues of A (λ = 2, 3) are positive. From this, we say that (0, 0) is an unstable node.
Thus, near (0, 0) we find:
FӼ3) /! Ӽ Ӽ Lņņl Z ӼbӼ Ӽ 9< V˅ Ӽ Ӽ *?GĖ"LĖ F t Ӽ/ )Ӽ
9͙
+'3/%6 8 5,„
Ӽ3Ӽ ///ӼӼӼ3)Ӽ+ӼLņņbӼªӼ SƒӼ 3 /ӼӼ
ĽņņUZ ³µӼZӼ,Ӽ ӼӼ î ‚ SӼU´ //ӼӼ3 /Ӽ!/ǶӼ Ӽ Ӽ
ZӼ t Ӽ Ӽ 3 /Ӽ Ӽ Ӽ /,Ӽ 3 ñӼ
,Ӽ Ӽ Ӽ 3 Ӽ ,Ӽ Ӽ - // Ӽ ēL› ÛRņF! ZӼ Ӽ
Ӽ Ӽ Ӽ 9< V˅ /\ Ӽ/\Ӽ3!Ӽ)CDC L Ė
UWZ bµ ėӼ FӼ &͙ ˅
B Ė
Œ˅
¦bӼ what you
4. Find the eigenvalues of the Jacobian matrix evaluated at each steady-state.¦bӼ
Using
know about classifying linear systems, classify each steady-state as a saddle, source, sink,
F Ӽ- ‚Ӽ Ӽ 3) /! Ӽ Lņ˅ + pĖ̴bӼªӼ Ӽ Ӽ Ӽ Ӽ+-Ӽ ªӼ Ӽ
spiral source, or spiral sink. (Note that if the eigenvalues are purely imaginary, or if one or
more eigenvalues are zero, then linear stability analysis is can be wrong. More work is needed
in these cases).
Œ p X “ b ¾ ŒX b ¾
. D G¾
5. Plot the location of each steady-state in the xy-plane. Using the classification obtained above,
sketch a mini phase-portrait around each steady-state. Finally, connect the trajectories as
expected to sketch the complete phase-plane diagram.
2