Transcript Differential Equations

Differential Equations
Chapter 07:
Nonlinear
Differential
Equations and
Stability
Brannan
Copyright © 2010 by John Wiley & Sons, Inc.
All rights reserved.
Chapter 7 Nonlinear Differential
Equations and Stability

In this chapter, we take up the
investigation of systems of nonlinear
equations. Such systems can be solved
by analytical methods only in rare
instances. Numerical approximation
methods provide one means of dealing
with nonlinear systems. Another
approach, presented in this chapter, is
geometrical in character and leads to a
qualitative understanding of the
behavior of solutions rather than to
detailed quantitative information.
Chapter 7 - Nonlinear Differential
Equations and Stability
 7.1 Autonomous
Systems and
Stability
 7.2 Almost Linear Systems
 7.3 Competing Species
 7.4 Predator–Prey Equations
 7.5 Periodic Solutions and Limit
Cycles
 7.6 Chaos and Strange Attractors:
The Lorenz Equations
7.1 Autonomous Systems and
Stability
We first introduced two-dimensional systems of the form
dx/dt = F(x, y), dy/dt = G(x, y)
(1)
in Section 3.6. Recall that the system (1) is called
autonomous because the functions F and G do not depend
on the independent variable t. In Chapter 3, we were mainly
concerned with showing how to find the solutions of
homogeneous linear systems, and we presented only a few
examples of nonlinear systems. Now we want to focus on the
analysis of two dimensional nonlinear systems of the form(1).
Unfortunately, it is only in exceptional cases that solutions can
be found by analytical methods. One alternative is to use
numerical methods to approximate solutions. Software
packages often include one or more algorithms, such as the
Runge–Kutta method discussed in Section 3.7, for this
purpose.
Nonlinear autonomous systems of
equations
We are concerned with systems of two
simultaneous differential equations
dx/dt = F(x, y), dy/dt = G(x, y)
(1)
where F and G are continuous and have
continuous partial derivatives in some
domain D of the xy-plane.
If (x0, y0) is a point in this domain, then there
exists a unique solution x = φ(t), y = ψ(t) of
the system (1) satisfying the initial
conditions x(t0) = x0, y(t0) = y0.
Stability and Instability.
Consider the autonomous systems of the
form x' = f(x).
 The points, if any, where f(x)=0 are called
critical points of the autonomous system.
 A critical point x0 of the system is said to be
stable if, given any ε > 0, there is a δ > 0
such that every solution x = φ(t) of the
system, which at t = 0 satisfies ||φ(0) − x0||
< δ, exists for all positive t and satisfies
||φ(t) − x0|| < ε for all t ≥ 0. A critical point
that is not stable is said to be unstable.

Asymptotically stable
The Oscillating Pendulum.

For the Oscillating
Pendulum, the
angular momentum
about the origin,
mL2(dθ/dt), is the
product of the
mass m, the
moment arm L, and
the velocity Ldθ/dt.
Thus the equation
of motion is
The Oscillating Pendulum.
By straightforward algebraic operations,
we can write this eq. in the standard
form
d2θ/dt2+ γdθ/dt+ ω2 sin θ = 0,
where γ = c/mL and ω2 = g/L.
 To convert this Eq. to a system of two first order
equations, we let x = θ and y = dθ/dt; then
dx/dt= y,
dy/dt = −ω2 sin x − γ y.
Since γ and ω2 are constants, the system is an
autonomous system of the form (1). The critical
points are found by solving the equations
y = 0, −ω2 sin x − γ y = 0.
We obtain y = 0 and x = ±nπ, where n is an integer.
The Importance of Critical Points.

Critical points correspond to equilibrium
solutions, that is, solutions in which x(t) and
y(t) are constant. For such a solution, the
system described by x and y is not
changing; it remains in its initial state
forever. It might seem reasonable to
conclude that such points are not very
interesting. However, recall that in Section
2.4 and later in Chapter 3, we found that the
behavior of solutions in the neighborhood of
a critical point has important implications for
the behavior of solutions farther away.
Examples - Undamped Pendulum
Example
7.2 Almost Linear Systems

TABLE 7.2.1
THEOREM 7.2.1
The critical point x=0 of the linear
system x' = Ax. is asymptotically
stable if the eigenvalues λ1, λ2 are real
and negative or are complex with
negative real part; stable, but not
asymptotically stable, if λ1 and λ2 are
pure imaginary; unstable if λ1 and λ2
are real and either is positive, or if
they are complex with positive real
part.
Effect of Small Perturbations.

The eigenvalues λ1,
λ2 are the roots of the
polynomial equation
det(A − λI) = 0. It is
possible to show that
small perturbations in
some or all of the
coefficients are
reflected in small
perturbations in the
eigenvalues.
Linear Approximations to Nonlinear
Systems.
Let us consider what it means for a nonlinear system
x‘=f(x) (3) to be “close” to a linear system (1).
Accordingly, suppose that x' = Ax + g(x) (4) and that
x = 0 is an isolated critical point of the system (4).
 This means that there is some circle about the origin
within which there are no other critical points. In
addition, we assume that det A = 0, so x = 0 is also
an isolated critical point of the linear system x' = Ax.
For the nonlinear system (4) to be close to the linear
system x = Ax, we must assume that g(x) is small.
More precisely, we assume that the components of g
have continuous first partial derivatives and satisfy
the limit condition
||g(x)||/||x||→0 as x → 0,
that is, ||g|| is small in comparison to ||x|| itself near the
origin. Such a system is called an almost linear
system in the neighborhood of the critical point x =
0.

Examples

1.

2.
Jacobian matrix
Let us now return to the general nonlinear system
which we write in the scalar form
x' = F(x, y), y' = G(x, y).
(10)
 We assume that (x0, y0) is an isolated critical point of
this system. The system (10) is almost linear in the
neighborhood of (x0, y0) whenever the functions F and
G have continuous partial derivatives up to order 2. To
show this, we use Taylor expansions about the point
(x0, y0) to write F(x, y) and G(x, y) in the form
F(x, y) = F(x0, y0) + Fx (x0, y0)(x − x0) + Fy (x0, y0)( y − y0)
+ η1(x, y),
G(x, y) = G(x0, y0) + Gx (x0, y0)(x − x0) + Gy (x0, y0)( y −
y0) + η2(x, y),
where {η1(x, y)/[(x − x0)2 + ( y − y0)2]1/2}→0 as (x, y)→(x0,
y0), and similarly for η2. Note that F(x0, y0) = G(x0, y0) =
0; also dx/dt = d(x − x0)/dt and dy/dt = d( y − y0)/dt.

Jacobian matrix

Then the system reduces to
or
where u1 = x − x0 and u2 = y − y0.
Jacobian matrix

The matrix
which appears as the coefficient matrix in above
equation is called the Jacobian matrix of the
functions F and G with respect to the variables
x and y. We need to assume that det(J) is not
zero at (x0, y0) so that this point is also an
isolated critical point of the linear system,
(13)
Example
THEOREM 7.2.2
Let λ1 and λ2 be the eigenvalues of the
linear system (1), x' = Ax, corresponding to
the almost linear system (4),
x' = Ax + g(x).
 Assume that x = 0 is an isolated critical
point of both of these systems. Then the
type and stability of x = 0 for the linear
system (1) and for the almost linear system
(4) are as shown in Table 7.2.2.

Table 7.2.2
Damped Pendulum.
Discuss the Damped Pendulum whose
characteristic equation is
λ2 + γλ + ω2 = 0,
3 cases
 1. If γ2 − 4ω2 > 0, then the eigenvalues are real,
unequal, and negative. The critical point (0, 0) is an
asymptotically stable node of the linear system and
of the almost linear system.
 2. If γ2 − 4ω2 = 0, then the eigenvalues are real,
equal, and negative. The critical point (0, 0) is an
asymptotically stable (proper or improper) node of
the linear system. It may be either an
asymptotically stable node or spiral point of the
almost linear system.
 3. If γ2 − 4ω2 < 0, then the eigenvalues are complex
with a negative real part.

7.3 Competing Species
Let x and y be the populations of the two
species at time t. Assume that the population
of each of the species, in the presence of the
other, is governed by a logistic equation.
dx/dt= x(ε1 − σ1x− α1 y),
dy/dt= y(ε2 − σ2 y− α2x),
respectively, where ε1 and ε2 are the growth
rates of the two populations, and ε1/σ1 and
ε2/σ2 are their saturation levels and where α1
is a measure of the degree to which species y
interferes with species x and α2 is a measure
to which species x interferes with species y.

Example
Example (Ctd.) - A phase portrait of
the system