# TWO DIMENSIONAL FLOWS Lecture Linear and Nonlinear Systems

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```					TWO DIMENSIONAL
FLOWS

Lecture 4: Linear and
Nonlinear Systems
4. Linear and Nonlinear Systems in 2D

In higher dimensions, trajectories have more
room to manoeuvre, and hence a wider range
of behaviour is possible.

4.1 Linear systems: deﬁnitions and examples

A 2-dimensional linear system has the form
˙
x = ax + by
˙
y = cx + dy
where a, b, c, d are parameters. Equivalently,
in vector notation
˙
x = Ax                   (1)
where
a b                   x
A=            and    x=             (2)
c d                   y
The Linear property means that if x1 and x2
are solutions, then so is c1x1 + c2x2 for any
c1 and c2.

The solutions of x = Ax can be visualized
˙
as trajectories moving on the (x, y) plane, or
phase plane.
1
Example 4.1.1 m¨ + kx = 0 i.e. the simple
x
harmonic oscillator

Fig. 4.1.1

The state of the system is characterized by
˙
x and v = x

˙
x = v
k
˙
v = −      x
m
˙ ˙
i.e. for each (x, v) we obtain a vector (x, v) ⇒
vector ﬁeld on the phase plane.

2
As for a 1-dimensional system, we imagine a
ﬂuid ﬂowing steadily on the phase plane with
a local velocity given by (x, v) = (v, −ω 2x).
˙ ˙

Fig. 4.1.2

• Trajectory is found by placing an imag-
inary particle or phase point at (x0, v0)
and watching how it moves.

• (x, v) = (0, 0) is a ﬁxed point:
static equilibrium!

• Trajectories form closed orbits around (0, 0):
oscillations!

3
The phase portrait looks like...

Fig. 4.1.3

• NB ω 2x2 + v 2 is constant on each ellipse.
This is simply the energy

Example 4.1.2

˙
x        a 0           x
=
˙
y        0 −1          y

4
The phase portraits for these uncoupled equa-
tions are...

Fig. 4.1.4

Solution is
x         x0eat
=
y         y0e−t

5
Some terminology...

• x∗ = 0 is an attracting ﬁxed point in Figs
(a) - (c) since x(t) → x∗ as t → ∞.

• x∗ = 0 is called Lyapunov Stable in Figs
(a) - (d) since all trajectories that start
suﬃciently close to x∗ remain close to it
for all time.

• Fig. (d) shows that a ﬁxed point can
be Lyapunov stable but not attracting ⇒
it is neutrally stable. It is also possible
for a ﬁxed point to be attracting but not
Lyapunov stable!

• If a ﬁxed point is both Lyapunov sta-
ble and attracting, we’ll call it stable, or
sometimes asymptotically stable

• x∗ is unstable in Fig. (e) because it is
neither attracting nor Lyapunov stable

6
4.2 Classiﬁcation of Linear Systems

Consider a general 2 ×2 matrix A such that
˙
x = Ax

To solve: try

x(t) = eλtv (v is a constant vector)
⇒ λeλt v = eλt Av
⇒ Av = λv
Hence if we obtain the eigenvectors v and
eigenvalues λ, we will have two independent
a b
solutions x(t). Recall that A =         has
c d
eigenvalues λ1 and λ2, where

τ+     τ 2 − 4∆          τ−   τ 2 − 4∆
λ1 =                     λ2 =
2                      2

with    τ = trace(A) = a + d
∆=    det(A)      = ad − bc

7
• Useful check when calculating eigenval-
ues: λ1 + λ2 = τ and λ1λ2 = ∆

˙
x          1 1    x
Example 4.2.1       =
˙
y          4 −2   y

1
• ⇒ λ1 = 2 with v1 =             λ1 > 0
1
hence solution grows

1
• ⇒ λ2 = −3 with v2 =                λ2 < 0
−4
hence solution decays

Fig. 4.2.1
8
• straight line trajectories in Fig. 4.2.1 are
the eigenvectors v1 and v2

Example 4.2.2 Consider λ2 < λ1 < 0

Fig. 4.2.2

• Both solutions decay exponentially!

9
Example 4.2.3 What happens if λ1, λ2 are
complex?

Fixed point is either...

Fig. 4.2.3

• If λ1, λ2 are purely imaginary, all solutions
are periodic

• If λ1 = λ2 we get a star node or a degen-
erate node

10
Classiﬁcation of Fixed Points

λ1,2 = 1 (τ ± τ 2 − 4∆),
2                 where
∆ = λ1λ2 and τ = λ1 + λ2

Fig. 4.2.4
11
4.3 Phase Portraits

˙
Recall x = f (x), i.e.

x˙1 = f1(x1, x2)
x˙2 = f2(x1, x2)
where x = (x1, x2) and f (x) = (f1(x), f2(x))
(not necessarily linear now). The trajectories
x(t) wind their way through the phase plane.

The entire phase plane is ﬁlled with trajec-
tories!

4.4 Example of a phase portrait

- Shows a sample of the qualitatively diﬀerent
trajectories
12
Fig. 4.4.1

• Fixed points A, B and C satisfy f (x∗ ) = 0
and correspond to steady states or equi-
libria

• Closed orbit D corresponds to periodic
solutions, i.e. x(t + T ) = x(t) for all t for
some T > 0

• The existence and uniqueness theorem
given for 1-dimensional systems can be
generalized to 2-dimensional systems ...
fortunately ⇒ diﬀerent trajectories never
intersect!

13
4.5 Fixed points and Linearization

This is the same idea as for 1-dimensional
systems
˙
x = f (x, y)
˙
y = g(x, y)
Suppose (x∗, y ∗) is a ﬁxed point. Expand
around (x∗, y ∗) using u = x−x∗ and v = y−y ∗.
u = x = f (x∗ + u, y ∗ + v)
˙   ˙
∂f     ∂f
∗ ∗
= f (x , y ) + u    +v     + O(u2, v 2, uv)
∂x     ∂y
∂f     ∂f
≃ u    +v
∂x     ∂y
Similarly
∂g     ∂g
˙
v≃u      +v
∂x     ∂y
Hence a small disturbance around (x∗, y ∗) evolves
as
∂f    ∂f
         
˙
u                    u
=  ∂x
∂g
∂y 
∂g
˙
v                    v
∂x    ∂y
where the matrix is known as the Jacobian
matrix A at (x∗, y ∗), and is the multivariable
equivalent of f ′ (x∗) for 1-D systems.
14
Example 4.5.1

x = −x + x3
˙
˙
y = −2y
˙
Fixed points occur where x = 0 and y =   ˙
0 simultaneously. Hence x = 0 or x = ±1
and y = 0 ⇒ 3 ﬁxed points (0, 0), (1, 0) and
(−1, 0)

Jacobian matrix A
˙
∂x    ˙
∂x
       
−1 + 3x2 0
A   =  ∂x
˙
∂y
∂y 
˙
∂y   =
0     −2
∂x   ∂y

−1 0
At (0, 0)       A=             ⇒ stable node
0 −2

2 0
At (±1, 0)      A=             ⇒ both are sad-
0 −2
dle points.

15
Fig. 4.5.1

In general, we must obtain ﬁxed points by
˙         ˙
solving x = 0 and y = 0 simultaneously.

e.g.   ˙
x = x(3 − x − 2y)
˙
y = y(2 − x − y)
yields ﬁxed points (0, 0), (0, 2), (3, 0) and
(1, 1)

In general, A will not be diagonal at (x∗, y ∗).
Hence we must diagonalize A, i.e. ﬁnd eigen-
values λ1 and λ2 and eigenvectors v1 and v2
of A
16
Basically, we are doing the same here as be-
fore for 2D linear systems, since we are treat-
ing the nonlinear system as linear near (x∗, y ∗).
Knowledge of λ1 and λ2, and v1 and v2, en-
ables us to sketch the phase portrait near
(x∗, y ∗).

The ﬁxed points can be classiﬁed according
to their stability as follows:

• If Re(λ1) > 0 and Re(λ2) > 0
⇒ repeller (unstable node)

• If Re(λ1) < 0 and Re(λ2) < 0
⇒ attractor (stable node)

• If Re(λ1) > 0 but Re(λ2) < 0 (or vice

• If λ1 and λ2 are both imaginary ⇒ centre

17
4.6 Example: Rabbits vs Sheep

An example of the Lotka-Volterra model of
competition between two species (e.g. rab-
bits and sheep) grazing the same food supply
(grass).

• Each species grows to its carrying capac-
ity in the absence of the other - logistic
growth (rabbits faster...!)

• When species encounter each other, the

• Conﬂicts occur at a rate proportional to
the size of each population. Conﬂicts re-
duce the growth rate of each species (but
more for rabbits).

A model encapsulating these properties could
be (see above!)
˙
x = x(3 − x − 2y)
˙
y = y(2 − x − y)
18
Fixed points at
3 0
(0, 0)   where    A =           ⇒ λ = 3, 2
0 2
−1 0
(0, 2)   where    A =               ⇒ λ = −1, −2
−2 −2
−1 −2                √
(1, 1)   where    A =               ⇒ λ = −1 ± 2
−1 −1
−3 −6
(3, 0)   where    A =               ⇒ λ = −3, −1
0 −1

(0, 0): λ = 3, 2 ⇒ unstable node (repeller)

λ = 2 ⇒ v = (0, 1)      “slow eigendirection′′
λ = 3 ⇒ v = (1, 0)      “fast eigendirection′′
General rule...

Trajectories are tangential to the slow
eigendirection (i.e. smallest |λ|) at a node
19
(0, 2): λ = −1, −2 ⇒ stable node (attrac-
tor)

• Once again... Trajectories are tangential
to the slow eigendirection at a node

• Here λ = −1 ⇒ v = (1, −2) is the slow
eigendirection.

√
(1, 1): λ = −1 ± 2 ⇒ saddle point

20
(3, 0): λ = −3, −1 ⇒ stable node (attrac-
tor)

Putting these together, the phase portrait
becomes....

Fig. 4.6.1

NB: You don’t really need to calculate the
eigenvectors to get the right shape!
21
Biological interpretation...

• In general, one species eventually drives
the other to extinction; which species even-
tually dominates depends on initial pop-
ulations x0 = (x0, y0)

• Basin of attraction of an attracting ﬁxed
point x∗ deﬁned as the set of initial con-
ditions x0 such that x → x∗ as t → ∞.

• In this case, basin boundary is the stable
manifold of the saddle node at (1, 1)

Fig. 4.6.2
22
4.7 Conservative Systems

˙
Consider x = f (x). A conserved quantity of
this system is a real-valued continuous func-
tion E(x) that is constant on trajectories i.e.
dE/dt = 0.

Example 4.7.1 m¨ = −dV (z)/dz = F (z)
z

˙
Take x = z and y = z ⇒

˙
x = y
1
˙
y =   F (x)
m
1 ˙
E(z) = 2 mz 2 +V (z) is the total energy, which
is constant
1
⇒ E(x) ≡        my 2 + V (x)
2
dE(x)
= 0
dt
since total energy is constant.

23
¨
Example 6.5.2 θ + sin θ = 0

e.g. undamped simple pendulum
˙
θ = ν
˙
ν = − sin θ
Fixed points at (θ ∗, ν ∗) = (kπ, 0)

0 1
(0, 0) :   A=              ⇒ λ = ±i ⇒ centre
−1 0
(oscillations)

1
Energy E(θ, ν) = 2 ν 2 − cos θ is conserved,
since
dE
˙         ˙     ¨
= ν ν + sin θ θ = ν[θ + sin θ] = 0
dt

0 1
(π, 0) :   A=            ⇒ λ = ±1 ⇒ saddle
1 0
24
Phase portrait becomes...

Fig. 4.7.1

25

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