CONTENTS
1
INTRODUCTION TO NONLINEAR DYNAMICAL SYSTEMS
Lecture 1: Introduction
Cameron Nowzari
August 25, 2015
Abstract
This lecture states the goals of the course and provides an introduction to basic concepts. We discuss
general dynamical properties of nonlinear systems and explore its differences with linear systems. We
also introduce some interesting nonlinear examples. The treatment corresponds to Chapters 1-2 in [2].
Contents
1 Introduction to nonlinear dynamical systems
1
1.1
Linear versus nonlinear
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Goals of this course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Basic concepts
3
3 Examples of nonlinear systems
4
3.1
Bowing of a violin string . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
3.2
The buckling beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
4 Planar dynamical systems
6
4.1
Linear systems on the plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
4.2
Linearization of nonlinear systems around an equilibrium point . . . . . . . . . . . . . . . . .
7
5 Closed orbits and limit cycles
9
5.1
Bendixson’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
5.2
Poincaré-Bendixson theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1
1.1
Introduction to nonlinear dynamical systems
Linear versus nonlinear
There is a huge body of work in the analysis and control of linear systems, which are of the form
ẋ(t) = A(t)x(t) + B(t)u(t) ,
x(0) = x0
1
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
1.1
Linear versus nonlinear
1
INTRODUCTION TO NONLINEAR DYNAMICAL SYSTEMS
where x ∈ Rn (the state), u ∈ Rm (the control inputs), A(t) ∈ Rn×n (the dynamics) and B(t) ∈ Rn×m (the
control action).
Most models currently available are linear - especially in industry. However, most real systems are nonlinear...
Why nonlinear theory? Because the dynamics of linear systems are not rich enough to describe many
commonly observed phenomena like
• Finite escape time. The state of an unstable linear system goes to infinity as time approaches infinity;
however, a nonlinear system’s state can go to infinity in finite time.
• Multiple (finite) equilibria. Consider the linear differential equation
ẋ(t) = A x(t)
(1)
For the initial condition x0 ∈ Rn , the solution of the system in x(t) = exp(At)x0 .
The point x = 0 is an equilibrium. If A is non-singular, then x = 0 is the only equilibrium of the
system. If A is singular, then ker A is the set of equilibria. However, an infinitesimal perturbation
“almost surely” makes any singular matrix non-singular.
Therefore, problems in nature and engineering which have a finite number of equilibria cannot be
described by means of a linear system model.
Example: a pendulum
• (Finite number of ) limit cycles. A limit cycle is a nontrivial periodic solution. Consider the
linear system (1). If A has eigenvalues in the imaginary axis, then the system admits a continuum of
periodic solutions. However, infinitesimal perturbations will cause the eigenvalues to be displaced of
the imaginary axis, and hence destroy the existence of periodic solutions.
Therefore, problems in nature and engineering which have a finite number of limit cycles cannot be
described by means of a linear system model.
Example: the pumping heart
• Bifurcations. Many real systems exhibit qualitative changes as the parameters of the model are varied.
These include changes in the number of equilibrium points, limit cycles and their stability properties.
However, linear systems are very simple in this regard. As the parameters of the model change, the
system can only change from stable to unstable (or vice versa). The number of equilibria goes to infinity
if one of the eigenvalues of the matrix A goes through the origin. Limit cycles only exist (and in an
infinite number) when the matrix A has eigenvalues in the imaginary axis.
Example: buckling of a beam
• Chaos. The dynamical behavior of linear systems is quite simple. The responses are sums of exponentials, with exponents given by the eigenvalues of the matrix A. They decay to zero or blow up as t goes
to infinity when there are no imaginary eigenvalues - independently of the initial condition! When A
has eigenvalues in the imaginary axis, the solutions neither decay nor blow up for any initial condition.
On the contrary, many physical systems exhibit very complex dynamical behavior. Small changes in
the initial condition can make the trajectories vastly differ over time.
Example: population models, climatic models, turbulent fluid flow models
• Many more reasons we will see along the course...
Linear techniques are good (and, in some cases, useful for the analysis of nonlinear systems!), but often times
not sufficient.
2
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
1.2
Goals of this course
1.2
2
BASIC CONCEPTS
Goals of this course
(i) To learn and use various notions and tools for the analysis and control of nonlinear systems.
(ii) To get a feeling and gain insight into the complexity of nonlinear systems.
(iii) To introduce a wide variety of interesting, inherently nonlinear examples.
2
Basic concepts
Basic notions from differentiable topology on Rn that by now you should be familiar with are the following:
• B(x, δ) denotes a ball centered at x of radius δ, i.e., B(x, δ) = {y ∈ Rn | ky − xk2 < δ}
• a neighborhood N of a point x ∈ Rn is a set such that there exists δ > 0 with B(x, δ) ⊂ N
• a limit point x ∈ Rn of a set S ⊂ Rn is a point such that every neighborhood of x contains a point of
S \ {x}
• a closed set S is a set that contains all its limit points. A set is closed if and only if it contains its
boundary
• an open set S is a set such that for any x ∈ S, there exists a neighborhood of x contained in S
• a bounded set S is such that there exists K > 0 such that kxk ≤ K for all x ∈ S
• a compact set S is a bounded and closed set.
Consider the dynamical system
ẋ = f (x, t),
where f : Rn × R → Rn . Let us define the following notions:
(i) the system is called autonomous if f does not explicitly depend on time, i.e., f (x, t) = f (x);
(ii) a point x0 ∈ Rn is called an equilibrium point if f (x0 , t) = 0 for all t;
(iii) an equilibrium x0 is called isolated if there exists δ > 0 such that there is no other equilibrium point in
the ball B(x0 , δ)
Let x0 be an equilibrium of the autonomous system ẋ = f (x). Assume f is C 1 . Writing down the Taylor
series about x0 , we get
f (x) = f (x0 ) +
∂f
∂f
(x0 )(x − x0 ) + r(x) =
(x0 )(x − x0 ) + r(x),
∂x
∂x
where r(x) is the tail of the Taylor series corresponding to the higher-order terms, i.e.,
lim
kx−x0 k2 →0
kr(x)k2
=0
kx − x0 k2
Denoting y = x − x0 , near x0 , we can approximate the nonlinear system by its linearization
ẏ =
∂f
(x0 )y.
∂x
The linearization of a nonlinear system tells us how the system behaves around x0 in some cases.
3
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
3
Theorem 2.1 If the Jacobian matrix
linear system.
∂f
∂x (x0 )
EXAMPLES OF NONLINEAR SYSTEMS
is non-singular, then x0 is an isolated equilibrium of the non-
Another situation when the linearization might be useful is when deciding the stability properties of an
equilibrium point. We will discuss this at depth in the chapter on Lyapunov stability – it is intrinsically
related to the following (famous) result.
Theorem 2.2 (Hartman-Grobman) If the Jacobian matrix ∂f
∂x (x0 ) has no eigenvalues in the imaginary
axis, then there exists δ > 0 and a continuous map h : B(x0 , δ) → Rn , with a continuous inverse (i.e., a
homeomorphism) mapping the trajectories of the nonlinear system onto the trajectories of the linearization.
3
3.1
Examples of nonlinear systems
Bowing of a violin string
The following example is the mechanical analog of the bowing of a violin string (see Figure 1(a)). Bowing is
modeled by the friction between the body and the conveyor belt. The conveyor belt is moving at a constant
velocity b. Consider the second-order system
2
10
1
5
0
-6
-4
-2
2
4
6
-5
-10
-15
(a) Mechanical analog
(b)
-1
-2
0
1
2
3
4
(c) Phase portrait
Figure 1: Bowing of a violin string.
M ẍ + kx + Fb (ẋ) = 0,
where the function Fb is of the form plotted in Figure 1(b). Defining x1 = x and x2 = ẋ, we can rewrite it
as a system of first-order differential equations
ẋ1 = x2
1
ẋ2 =
− kx1 − Fb (x2 )
M
The equilibrium points of the system are determined by
x2 = 0,
−kx1 − Fb (x2 ) = 0
Therefore, there is only one equilibrium point, x0 = (− Fbk(0) , 0), which is obviously isolated. To assess its
stability properties, we linearize the system around it. The Jacobian at x0 is
0
1
∂f
0
(x0 ) =
F (0)
k
∂x
−M
− bM
4
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
3.2
The buckling beam
3
EXAMPLES OF NONLINEAR SYSTEMS
For instance, for k = 3, M = 3 and Fb defined by
(
−((z − b) + c)2 − d,
Fb (z) =
((z − b) − c)2 − d,
x<b
x>b
with b = 1, c = 2 and d = 3, we get x0 = ( 34 , 0) and
∂f
0
(x0 ) =
−1
∂x
which has eigenvalues
3.2
1
3
±
1
2
3
√
2 2
3 i.
The buckling beam
Consider the following model
mẍ + dẋ − µx + λx + x3 = 0
Here x represents the one-dimensional deflection of the beam normal to the axial direction, µx stands for the
applied axial force, λx + x3 models the restoring spring force in the beam, and dẋ is the damping. We can
rewrite the system as
ẋ1 = x2
ẋ2 = −
1
dx2 − µx1 + λx1 + x31
m
The equilibrium points are determined by
x2 = 0,
dx2 − µx1 + λx1 + x31 = 0
√
Therefore, for λ ≤ µ, the system has three equilibrium points (0, 0), (± µ − λ, 0). For λ > µ, the only
equilibrium point is (0, 0). Figure 2 shows a phase portrait for the undamped case. The linearization of the
system is
∂f
0
1
(x) = 1
d
2
−m
∂x
m µ − λ − 3x1
For the values d = 1, m = 1, µ = 2 and λ = 1, we get the equilibrium points (0, 0), (−1, 0), and (1, 0). The
corresponding eigenvalues of the linearization at (0, 0) are
√
−1 ± 5
2
and at (±1, 0),
√
1
7
− ±
i
2
2
Therefore, (0, 0) is unstable and the other two equilibria are stable.
5
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
4
x’=y
y ’ = − (d y − mu x + lambda x + x3)/m
PLANAR DYNAMICAL SYSTEMS
d=1
mu = 2
m=1
lambda = 1
1
0.8
0.6
0.4
y
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
Figure 2: Phase portrait of the buckling beam for d = 1, m = 1, µ = 2 and λ = 1.
4
Planar dynamical systems
A planar dynamical system is represented by two scalar differential equations
ẋ1 = f1 (x1 , x2 )
ẋ2 = f2 (x1 , x2 )
(2)
The (x1 , x2 )-plane is usually called the phase plane. The ‘bowing of a violin string’ and the ‘buckling beam’
above are both planar examples.
If t 7→ x(t) = (x1 (t), x2 (t)) ∈ R2 denotes the solution of the system that starts at x0 ∈ R2 , the velocity vector
of this curve in the (x1 , x2 )-plane at time t is precisely the vector
(ẋ1 (t), ẋ2 (t)) = f (x1 (t), x2 (t)) = (f1 (x1 (t), x2 (t)), f2 (x1 (t), x2 (t))).
The vector field x ∈ R2 → f (x) ∈ R2 can be plotted with Mathematica/Matlab/Maple to give us an idea of
the behavior of the solutions of the system. In Mathematica, use the command PlotVectorField. In Matlab,
use the routine pplane provided by Prof. Polking at Rice University at http://math.rice.edu/~dfield
Example 4.1 (Tunnel-diode circuit) Consider the following equations describing the behavior of a tunneldiode circuit
ẋ1 = .5(−h(x1 ) + x2 )
ẋ2 = .2(−x1 − 1.5x2 + 1.2)
2
with h(x) = 17.76x − 103.79x + 229.62x3 − 226.31x4 + 83.72x5 . Solving for the equilibrium points, we get
Q1 = (.063, .758),
Q2 = (.285, .61),
Q3 = (.884, .21).
•
Figure 3 shows the phase portrait of this system.
6
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
4.1
Linear systems on the plane
4
PLANAR DYNAMICAL SYSTEMS
x ’ = .5 ( − 17.76 x + 103.79 x2 − 229.62 x3 + 226.31 x4 − 83.72 x5 + y)
y ’ = .2 ( − x − 1.5 y + 1.2)
1.6
1.4
1.2
1
y
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.4
−0.2
0
0.2
0.4
0.6
x
0.8
1
1.2
1.4
1.6
Figure 3: Phase portrait of the tunnel-diode circuit example.
4.1
Linear systems on the plane
Consider the linear time-invariant system
ẋ = Ax
(3)
where A is a 2 × 2 matrix. From linear algebra, we know that there exits an invertible matrix Q such that
Q−1 AQ = J, where J may take one of the following forms
λ1 0
λ 0
λ 1
α −β
,
,
,
.
(4)
0 λ2
0 λ
0 λ
β α
The solution of (3) starting from x0 is given by x(t) = Q exp(Jt)Q−1 x0 . The first case (4) corresponds to
the case when the eigenvalues λ1 and λ2 are real and distinct. The second and third case corresponds to the
case when the eigenvalues are real and equal (second case, two linearly independent eigenvectors, third case
only one). Finally, the fourth case corresponds to the case of complex eigenvalues α ± iβ.
(Un)stable (improper) node, saddle point, (un)stable focus, center: follow discussion in [2, Chapter 2.1]
4.2
Linearization of nonlinear systems around an equilibrium point
Consider the linearization of the nonlinear system (2) around an equilibrium point x0 ∈ R2 ,
ẏ = Ay
where y ∈ R2 and
A=
a11
a21
∂f
a12
=
(x0 ).
a22
∂x
Recall the Hartman-Grobman theorem, cf. Theorem 2.2. From our previous discussion on linear systems, we
deduce
7
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
4.2
Linearization of nonlinear systems around an equilibrium point4
PLANAR DYNAMICAL SYSTEMS
(i) if the origin of the linearization is a stable (respectively unstable) node with distinct eigenvalues, then
the equilibrium of the nonlinear system is a stable (respectively unstable) node;
(ii) if the origin of the linearization is a stable (respectively unstable) focus, then the equilibrium of the
nonlinear system is a stable (respectively unstable) focus;
(iii) if the origin of the linearization is a saddle point, then the equilibrium of the nonlinear system is a
saddle point.
If the origin of the linearization is a center point, then we cannot make any claims about the nonlinear system.
Example 4.2 (Tunnel-diode circuit revisited) Consider again the tunnel-diode circuit example. The
Jacobian matrix of the system is
∂f
−.5h0 (x1 ) .5
=
−.2
−.3
∂x
Evaluating this at each of the equilibrium points, we get
∂f
−3.598 .5
(Q1 ) =
, eigenvalues −3.57, −.33 ⇒ stable node
−.2
−.3
∂x
∂f
1.82 .5
(Q2 ) =
, eigenvalues 1.77, −.25 ⇒ saddle point
−.2 −.3
∂x
∂f
−1.427 .5
(Q3 ) =
, eigenvalues −1.33, −.4 ⇒ stable node
−.2
−.3
∂x
•
Figure 4 shows various solutions of this system.
x ’ = .5 ( − 17.76 x + 103.79 x2 − 229.62 x3 + 226.31 x4 − 83.72 x5 + y)
y ’ = .2 ( − x − 1.5 y + 1.2)
1.6
1.4
1.2
1
y
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.4
−0.2
0
0.2
0.4
0.6
x
0.8
1
1.2
1.4
1.6
Figure 4: Phase portrait of the tunnel-diode circuit example.
Example 4.3 Consider the dynamical system
ẋ1 = x2
ẋ2 = −x1 − εx21 x2
8
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
5
CLOSED ORBITS AND LIMIT CYCLES
The linearization around the equilibrium point (0, 0) is
ẏ1 = y2
ẏ2 = −y1
with eigenvalues ±i. Therefore, it is inconclusive about the stability properties of the equilibrium point of
the nonlinear system (we cannot apply the Hartman-Grobman theorem). In fact, checking out the evolution
of the function V (x1 , x2 ) = 12 (x21 + x22 ) along the solutions of the system, we get
d
(V (x1 (t), x2 (t))) = −εx21 x22 .
dt
Therefore, for ε > 0, the equilibrium is stable, and for ε < 0 it is unstable (see Figure 5). Note how the
linearization around the equilibrium is completely blind to this fact, i.e., no matter the value of ε, it always
gives the same answer (“the equilibrium of the linear system is a center”). We will learn how to invoke this
kind of argument systematically when we discuss Lyapunov stability analysis.
•
eps = 1
x’=y
y ’ = − x − eps x2 y
2
1.5
1.5
1
1
0.5
0.5
0
y
y
x’=y
2
y ’ = − x − eps x y
2
−0.5
eps = − 1
0
−0.5
−1
−1
−1.5
−1.5
−2
−2
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
−2
−1.5
−1
−0.5
0
x
0.5
1
1.5
2
Figure 5: Phase portraits for ε = 1 and ε = −1.
Example 4.4 The dynamical systems
ẋ = x3 ,
ẋ = −x3
have the same linearization around 0, simply ẏ = 0. However, their behaviors are very different.
5
•
Closed orbits and limit cycles
A non-trivial periodic solution of (2) satisfies x(t + T ) = x(t) for all t ≥ 0 and some period T > 0. Non-trivial
excludes the possibility of constant solutions corresponding to equilibrium points. A closed orbit γ is the
trace of the trajectory of a nontrivial periodic solution. An isolated closed orbit is called a limit cycle. All
limit cycles are closed orbits, but not all closed orbits are limit cycles.
Example 5.1 (Duffing’s equation) Consider the following dynamical system
ẍ + δ ẋ − x + x3 = 0
9
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
5.1
Bendixson’s theorem
5
CLOSED ORBITS AND LIMIT CYCLES
Writing this as a system of first-order differential equations, we get
ẋ1 = x2
ẋ2 = x1 − x31 − δx2
•
Figure 6(a) shows the phase portrait for δ = 0. Note the infinitely many number of closed orbits.
4
4
2
2
0
0
-2
-2
-4
-4
-2
-1
0
1
2
-2
(a) δ = 0
-1
0
1
2
(b) δ = 1
Figure 6: Phase portraits of the Duffing example.
5.1
Bendixson’s theorem
Theorem 5.2 (Bendixson’s theorem) Let D ⊂ R2 be a simply connected region. Consider the vector
field f : R2 → R2 and assume that its divergence
div(f ) =
∂f1
∂f2
+
∂x1
∂x2
is not identically zero and does not change sign on D. Then the dynamical system ẋ = f (x) does not possess
any closed orbits.
have
Proof: Let us reason by contradiction. Assume there exists a closed orbit γ. Along any orbit of f , we
dx2
dx1 = f2 /f1 . Therefore,
Z
f2 (x1 , x2 )dx1 − f1 (x1 , x2 )dx2 = 0
γ
By Green’s theorem, this implies that
Z Z ∂f1
∂f2 +
dx1 dx2 = 0
∂x2
S ∂x1
where S is the interior of γ. This contradicts the hypothesis that div(f ) is not identically zero and does not
change sign on D, and this concludes the proof.
Example 5.3 (Duffing’s equation revisited) Writing Duffing’s equation (cf. Example 5.1) as a system
of first-order differential equations, we get
ẋ1 = x2 ,
ẋ2 = x1 − x31 − δx2 ,
10
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
5.2
Poincaré-Bendixson theorem
5
CLOSED ORBITS AND LIMIT CYCLES
and recognize f (x1 , x2 ) = (x2 , x1 − x31 − δx2 ). We compute div(f ) = −δ. So for δ 6= 0, there are no closed
orbits on R2 . Figure 6(b) shows the phase portrait for δ = 1. Note that the closed orbits of Figure 6(a) have
all disappeared.
•
5.2
Poincaré-Bendixson theorem
Definition 5.4 (Positively invariant sets) A set M is called positively invariant for the dynamical system (2) if every trajectory starting in M stays in M for all future time.
Theorem 5.5 (Poincaré-Bendixson theorem) Let M ⊂ R2 be a non-empty, compact, positively invariant set. Then M contains an equilibrium point or a closed orbit.
The intuition behind this result is the trajectories in the set M will have to approach closed orbits or
equilibrium points as time tends to infinity.
Remark 5.6 (How to find invariant sets) Consider a simple closed curve defined by the equation V (x) =
c, where V : Rn → R is continuously differentiable. The derivative of V along the flow field f (Lie derivative)
is then given by
∇V · f =
∂V
∂V
f1 (x) +
f2 (x)
∂x1
∂x2
If this is less than or equal to zero at each point in the curve defined by V (x) = c, that means that the
vector field f is pointing inward or is tangent to the curve, and therefore, the trajectories
of the dynamical
system are trapped inside the closed region defined by the curve. The set M = x ∈ R2 | V (x) ≤ c is then
positively invariant with respect to f .
•
Example 5.7 (Harmonic oscillator) Consider the dynamical system
ẋ1 = x2
ẋ2 = −x1
and the annular region M = x ∈ R2 | c1 ≤ x21 + x22 ≤ c2 , with c2 > c1 > 0. The set M is closed, bounded,
and does not contain any equilibrium point. Moreover, since f (x) · ∇V (x) = 0 everywhere, trajectories are
trapped inside M , i.e., the set is positively invariant. Hence, by Poincaré-Bendixson theorem, we know that
there is at least one periodic orbit in M .
•
Example 5.8 (Lotka-Volterra predator-prey model with limited growth) Motivated by the study
of the cyclic variation of the populations of small fish in the Adriatic, Count Vito Volterra proposed the
following model for the population evolution of two species: prey x and predators y,
ẋ = (a − by − λx)x,
ẏ = (cx − d − µy)y,
where a, b, c, d, λ, µ > 0. The equilibria of the system are
e1 = (0, 0),
a
e2 = ( , 0),
λ
e3 = (
bd + aµ ac − dλ
,
),
bc + λµ bc + λµ
d
e4 = (0, − )
µ
We neglect the last one, since our attention is restricted to positive values x, y ≥ 0 for the populations. For
the same reason, the third one is not considered if ac − dλ < 0. The Jacobian of the vector field takes the
form
∂f
a − by − 2λx
−bx
=
cy
cx − d − 2µy
∂x
11
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
5.2
Poincaré-Bendixson theorem
5
CLOSED ORBITS AND LIMIT CYCLES
From here, we get
∂f
a 0
(e1 ) =
, eigenvalues are a, −d ⇒ saddle point
0 −d
∂x
ac
∂f
a − ab
λ
(e2 ) =
, eigenvalues are −a,
− d ⇒ saddle point or stable node
0 ca
−
d
∂x
λ
λ
∂f
(e3 ) = . . .
∂x
Consider the following two cases
Case ac < dλ: In this case, we only have the equilibria e1 and e2 . The first one is a saddle point and the
second one is a stable node. A sample phase portrait is plotted in Figure 7. Note that the region A ∪ B
a=2
d=2
x ’ = (a − b y − lambda x) x
y ’ = (c x − d − mu y) y
c=1
b = .5
lambda = 2 mu = 2
6
5
y
4
3
a - by - λx = 0
2
B
cx-d-μy=0
1
0
A
C
0
1
2
3
x
4
5
6
Figure 7: Phase portrait of Lotka-Volterra model for ac < dλ
is non-empty, compact, and positively invariant. By Poincaré-Bendixson, this region must contain an
equilibrium point or a closed orbit. Note that ẏ ≤ 0 in this region, therefore there are no closed orbits.
Also, the trajectories starting in C eventually reach B (why?). Therefore, all trajectories eventually
converge to either e1 or e2 .
Case ac > dλ: In this case, we have the equilibria e1 , e2 and e3 . The first two ones are saddle points and the
third one is a stable focus/node (check!). A sample phase portrait is plotted in Figure 8. The region
plotted in the figure is non-empty, closed, bounded and positively invariant. By Poincaré-Bendixson,
it contains an equilibrium point or a closed orbit. By index theory, if such an orbit exists, it must
enclose the equilibrium e3 . All trajectories starting outside this region eventually enter it. Therefore,
all trajectories will eventually converge to the equilibria e1 , e2 , e3 , or the limit cycle if it exists.
This example shows how, without actually computing the explicit solutions of a dynamical system, we can
deduce a lot of things about its behavior with the qualitative tools that we have introduced.
•
12
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.
REFERENCES
REFERENCES
a=2
d=2
x ’ = (a − b y − lambda x) x
y ’ = (c x − d − mu y) y
c=1
b = .5
lambda = .5 mu = .5
6
5
cx-d-μy=0
4
y
a - by - λx = 0
3
2
1
0
0
1
2
3
x
4
5
6
Figure 8: Phase portrait of Lotka-Volterra model for ac > dλ
References
[1] S. S. Sastry, Nonlinear Systems: Analysis, Stability and Control, ser. Interdisciplinary Applied Mathematics.
Springer, 1999, no. 10.
[2] H. K. Khalil, Nonlinear Systems, 3rd ed.
Prentice Hall, 2002.
[3] D. V. Zenkov, A. M. Bloch, and J. E. Marsden, “The energy-momentum method for the stability of nonholonomic
systems,” Dynamics and Stability of Systems, vol. 13, pp. 123–166, 1998.
[4] S. H. Strogatz, Nonlinear Dynamics and Chaos.
Cambridge: Westview Press, 1994.
13
ESE 617 – Nonlinear Systems and Control
These notes have been adapted from Professor Jorge Cortés’ notes for MAE 281A at the University of California, San Diego.