Ch 9.3: Almost Linear SystemsIn Section 9.1 we gave an informal description of the stability properties of the equilibrium solution x = 0 of the 2 x 2 system x' = Ax. The results are summarized in Table 9.1.1.We required detA 0, and hence x = 0 is the only critical point of the system x' = Ax. Now that we have precisely defined the concepts of asymptotic stability, stability, and instability, we can restate these results in the Theorem 9.3.1, on the next slide.

Theorem 9.3.1The critical point x = 0 of the 2 x 2 linear system x' = Ax is (1) asymptotically stable if the eigenvalues r1 and r2 are real and negative or have negative real part; (2) stable, but not asymptotically stable, if r1 and r2 are pure imaginary; (3) unstable if r1 and r2 are real and either is positive, or if they have positive real part.

PerturbationsThus by Theorem 9.3.1, or Table 9.1.1, the eigenvalues r1, r2 of A determine the type of critical point at x = 0 and its stability characteristics. Now r1, r2 depend on the coefficients in the system x' = Ax, which in turn may depend on physical measurements. Since these measurements are typically subject to small uncertainties, it is of interest to investigate whether the small changes (perturbations) in the coefficients can affect the stability or instability of a critical point and/or significantly alter the pattern of the trajectories.

Perturbations in Pure Imaginary EigenvaluesRecall that the eigenvalues r1 and r2 of A are the roots of the polynomial equation det(A-rI) = 0. It can be shown that small perturbations in some or all of the coefficients are reflected in small changes in the eigenvalues. The most sensitive situation occurs when r1 = i and r2 = - i, that is, when the critical point is a center. Small changes to the coefficients results in r1, r2 taking on new values: r1 = ' + i' and r2 = ' - i', where ' 0 and ' .

Perturbations and CentersThus r1, r2 become r1 = ' + i' and r2 = ' - i'.

If ' 0, which almost always happens, then trajectories of perturbed system are spirals rather than closed curves. The system is asymptotically stable if ' < 0; unstable if ' > 0.Thus small perturbations in coefficients may change a stable system into an unstable one, and in any case may be expected to alter radically the trajectories in the phase plane.

Perturbations in Equal EigenvaluesAnother slightly less sensitive case occurs when r1 = r2, that is, when the critical point is a node. Small perturbations in the coefficients will normally cause the two equal roots to separate (bifurcate).If the separated roots are real, then the critical point remains a node, but if the separated roots are complex conjugates, then the critical point becomes a spiral point. Here, the stability or instability of the system is not affected by small changes in the coefficients, but the trajectories may be altered considerably.

Perturbations in Other CasesIn all other cases the stability or instability of the system is not changed, nor is the type of critical point altered, by sufficiently small perturbations in the coefficients of the system. For example, if r1 and r2 are real, negative, and unequal, then a small change in the coefficients will neither change the sign of r1 and r2 nor allow them to coalesce. Thus the critical point remains an asymptotically stable node.

Nonlinear SystemsConsider a nonlinear two-dimensional autonomous system

x' = f(x)Our main object is to investigate the behavior of trajectories of this system near a critical point x0.We do this by approximating the nonlinear system near a critical point x0 by an appropriate linear system, whose trajectories are easy to describe. The crucial question is whether the trajectories of the linear system are good approximations to those of nonlinear system.For convenience, assume critical point is at the origin, x0 = 0. This involves no loss of generality, since in general the substitution u = x – x0 shifts the critical point to the origin.

Nonlinear and Nearby Linear SystemsFirst, we consider what it means for a nonlinear system to be “close” to a linear system. Accordingly, suppose that x' = f(x) = Ax + g(x).Assume that x = 0 is an isolated critical point of this system. This means that there is some circle about the origin within which there are no other critical points.In addition, assume that detA 0, and hence x = 0 is also an isolated critical point of the system x' = Ax. For the nonlinear system x' = f(x) to be close to the linear system x' = Ax, we must assume that g(x) is small.

Almost Linear Systems: Vector FormRecall: x' = f(x) = Ax + g(x), where g(x) is small. More precisely, assume the components of g have continuous first partial derivatives and satisfy the limit condition

Thus ||g(x)|| is small compared to ||x|| near the origin. Such a system is called an almost linear system in the neighborhood of the critical point x = 0.

0)(

lim x

xg0x

Almost Linear Systems: Scalar FormRecall: x' = f(x) = Ax + g(x), where

In scalar form, if we let

then

and the corresponding limit condition is

0)(

lim x

xg0x

),(),(

)(,2

1

yxgyxg

yx

xgx

,),(),()(, 22

21

22 yxgyxgryx xgx

0),(limand0),(lim 2

0

1

0

ryxg

ryxg

rr

Example 1: Nonlinear System (1 of 2)

Consider the system

Note that (0, 0) is a critical point and detA 0. It can be shown that the other critical points are (0, 2), (1, 0) and (0.5, 0.5). Thus (0, 0) is an isolated critical point. The limit condition

is more readily verified if we use polar coordinates x = rcos and y = rsin .

2

2

25.075.05.0001

yxyxyx

yx

yx

0)(

lim x

xg0x

Example 1: Almost Linear System (2 of 2)

Using polar coordinates x = rcos and y = rsin, we have

Thus

and hence our system is almost linear near the origin.

2

22222

2

22221

sin25.0sincos75.0

sin25.0sincos75.025.075.0),(

,sincoscos

sincoscos),(

rr

rrr

yxyr

yxg

rrrr

rxyx

ryxg

0),(limand0),(lim 2

0

1

0

ryxg

ryxg

rr

Example 2: Pendulum SystemThe motion of a pendulum system is given by

or

Note detA 0, and critical points are (n , 0) for n = 1, 2,…. Thus (0, 0) is an isolated critical point, with g1(x,y) = 0 and

where, from the Taylor series representation of sinx,

Thus this system is almost linear.

yxyyx sin, 2

xxyx

yx

sin010 2

2

,0coslim)(sinlim),(lim332

0

2

0

2

0

rr

rxx

ryxg

rrr

52

31sin xaxaxx

General Nonlinear System (1 of 3)

Consider the general nonlinear system x' = f(x), or

We will show that this system is almost linear near a critical point (x0, y0) whenever the functions F and G have continuous partial derivatives up to order 2. To see this, we use Taylor expansions about the point (x0, y0) to write F(x,y) and G(x,y) in the form

where

),(),,( yxGyyxFx

),())(,())(,(),(),(

),,())(,())(,(),(),(

200000000

100000000

yxyyyxGxxyxGyxGyxG

yxyyyxFxxyxFyxFyxF

yx

yx

0)()(

),(lim

)()(

),(lim

20

20

2

),(),(20

20

1

),(),( 0000

yyxx

yx

yyxx

yxyxyxyxyx

Rewriting General Nonlinear System (2 of 3)

We have F(x,y) and G(x,y) in the form

Since (x0, y0) is a critical point, F(x0, y0) = G(x0, y0) = 0. Also, note that dx /dt = d(x - x0)/dt and dy /dt = d(y - y0)/dt.

Thus the original system of equations

reduces to

),())(,())(,(),(),(

),,())(,())(,(),(),(

200000000

100000000

yxyyyxGxxyxGyxGyxG

yxyyyxFxxyxFyxFyxF

yx

yx

),(),,( yxGyyxFx

),(),(

),(),(),(),(

2

1

0

0

0000

0000

0

0

yxyx

yyxx

yxGyxGyxFyxF

yyxx

dtd

yx

yx

General Nonlinear System: Almost Linear (3 of 3)

Thus our system of equations can be written as

In vector notation,

Thus if F and G are twice differentiable, then the nonlinear system of equations x' = f(x) is almost linear, and the linear system that approximates the nonlinear system is given by

),(),(

),(),(),(),(

2

1

0

0

0000

0000

0

0

yxyx

yyxx

yxGyxGyxFyxF

yyxx

dtd

yx

yx

),(),(

,where,2

1

0

00

yxyx

yyxx

dd

dtd

xηxuxηuxxfu

0

0

2

1

2

1

0000

0000

2

1 where,),(),(),(),(

yyxx

uu

uu

yxGyxGyxFyxF

uu

dtd

yx

yx

Example 3: Pendulum System (1 of 2)

The motion of a pendulum system is given by

Thus

The critical points are (n , 0) for n = 0, 1, 2,…. Since F and G are twice differentiable, the system of equations is almost linear near each critical point. We have

To find the approximating linear system at (x0, y0), we use

yxyyx sin, 2

yxyxGyyxF sin),(,),( 2

yxyx GxGFF ,cos,1,0 2

0

0

2

1

2

1

0000

0000

2

1 where,),(),(),(),(

yyxx

uu

uu

yxGyxGyxFyxF

uu

dtd

yx

yx

Example 3: Approximating Linear Systems (2 of 2)

To find the approximating linear system at (x0, y0), we use

with

At the origin, the approximating linear system is

At ( , 0), the approximating linear system is

yx

yx

dtd

2

10

0

0

2

1

2

1

0000

0000

2

1 where,),(),(),(),(

yyxx

uu

uu

yxGyxGyxFyxF

uu

dtd

yx

yx

y

xy

xdtd

2

10

yxyx GxGFF ,cos,1,0 2

Theorem 9.3.2Consider the almost linear system x' = Ax + g(x). Let r1 and r2 be the eigenvalues of A. Then the type and stability of the critical point (0,0) of the linear system x' = Ax and the almost linear system x' = Ax + g(x) are as given in the table below.

L i n e a r S y s t e m A l m o s t L i n e a r S y s t e m E i g e n v a l u e s T y p e S t a b i l i t y T y p e S t a b i l i t y

021 rr N o d e U n s t a b l e N o d e U n s t a b l e

021 rr N o d e A s y m p t o t i c a l l y S t a b l e

N o d e A s y m p t o t i c a l l y S t a b l e

12 0 rr S a d d l e P o i n t U n s t a b l e S a d d l e P o i n t U n s t a b l e

021 rr P r o p e r o r I m p r o p e r N o d e

U n s t a b l e N o d e o r S p i r a l P o i n t

U n s t a b l e

021 rr P r o p e r o r I m p r o p e r N o d e

A s y m p t o t i c a l l y S t a b l e

N o d e o r S p i r a l P o i n t

A s y m p t o t i c a l l y S t a b l e

irr 21 ,

0 S p i r a l P o i n t U n s t a b l e S p i r a l P o i n t U n s t a b l e 0 S p i r a l P o i n t A s y m p t o t i c a l l y

S t a b l e S p i r a l P o i n t A s y m p t o t i c a l l y

S t a b l e irir 21 , C e n t e r S t a b l e C e n t e r o r

S p i r a l P o i n t I n d e t e r m i n a t e

Theorem 9.3.2 Discussion (1 of 2)

Since nonlinear term g(x) is small compared to the linear term Ax when x is small, we hope that the trajectories of the linear system x' = Ax are good approximations to those of nonlinear system, at least near the origin. Theorem 9.3.2 states that this is true in many but not all cases. For small x, the nonlinear terms are small and do not affect the stability and type of critical point as determined by linear term, except in two sensitive cases: r1 and r2 pure imaginary, and r1 and r2 real and equal.

As we have seen earlier, small changes in eigenvalues can alter the type and stability of the critical point for a linear system, but only in these two cases. Thus a small nonlinear term might have a similar effect for these two cases as well.

Theorem 9.3.2 Discussion (2 of 2)

Even if the critical point is of the same type as that of the linear system, the trajectories of the almost linear system may be much different in appearance than for the linear system, except very near the critical point. However, it can be shown that the slopes at which the trajectories “enter” or “leave” the critical point are given correctly by the linear equation.

Damped Pendulum System (1 of 2)

Recall that the motion of a pendulum system is given by

Near the origin these nonlinear equations are approximated by

whose eigenvalues are

The nature of the solutions to the linear and nonlinear systems depends on the sign of 2 – 42, as examined on the next slide.

yxyyx sin, 2

yx

yx

2

10

24

,22

21

rr

Damped Pendulum System (2 of 2)

Thus we have the following cases:If 2 – 4 2 > 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.If 2 – 4 2 = 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 a spiral point of the almost linear system.If 2 – 4 2 < 0, then the eigenvalues are complex with negative real part. The critical point (0,0) is an asymptotically stable spiral point of the linear system, and of the almost linear system.

Hence the origin is a spiral point of the nonlinear system if the damping is small, and a node if damping is large enough. In either case, the origin is asymptotically stable.

24

,22

21

rr

Pendulum System: Small Damping (1 of 5)

Consider the small damping case, 2 – 4 2 < 0. The direction of motion of the spirals near (0,0) can be obtained directly from the equations

For the point at which a spiral intersects the positive y-axis, at x = 0 and y > 0, it follows that dx/dt > 0. Thus the point (x,y) on the trajectory is moving to the right, so the direction of motion spirals clockwise.

yxdtdyydtdx sin/,/ 2

Pendulum System: Small Damping (2 of 5)

The direction of motion of the spirals near (2n , 0) is the same as near the origin.As before, this can be obtained directly from the equations

We can expect this on physical grounds, since all these critical points correspond to the downward equilibrium position of the pendulum.

yxdtdyydtdx sin/,/ 2

Pendulum System: Small Damping (3 of 5)

Next, consider the critical point (, 0). Here, the nonlinear equations are approximated by the linear system

whose eigenvalues are

One eigenvalue (r1) is positive and the other (r2) is negative.

Therefore, regardless of the amount of damping, the critical point ( , 0) is an unstable saddle point both of the linear system and of the almost linear system.

,10

2

y

xy

xdtd

24

,22

21

rr

Pendulum System: Small Damping (4 of 5)

To examine the trajectories near the saddle point (, 0) in more detail, consider the general solution of the linear system:

Since r1 > 0 and r2 < 0, it follows that the solution that approaches zero as t corresponds to C1 = 0.

For this solution v/u = r2, thus slope of entering trajectories is negative; one lies in second quadrant (C2 < 0) and the other lies in the fourth quadrant (C2 > 0).

For C2 = 0, we obtain a pair of trajectories exiting from the saddle point. They have slope r1 > 0; one lies in first quadrant (C1 > 0) and the other lies in the third quadrant (C1 < 0).

y

xvu

er

Cer

Cvu trtr

where,11

21

22

11

Pendulum System: Small Damping (5 of 5)

The analysis at (, 0) can be repeated to show that the critical points (n, 0), n odd, are all saddle points oriented in the same way as the one at (, 0). These critical points all correspond to the upward equilibrium position of the pendulum, so we expect them to be unstable. Diagrams showing the trajectories in the neighborhood of two saddles points are given below.

Example 4: Pendulum System (1 of 6)

The motion of a certain pendulum system is given by

where x = and y = d /dt. Note that 2 = 9, and thus the damping coefficient, = 1/5, is relatively small. It follows that 2 – 4 2 < 0 here. The phase portrait for this system is given below.

5/sin9, yxyyx

Example 4: Critical Points (2 of 6)

The critical points are (n , 0) for n = 0, 1, 2,…. Even values of n, including (0,0), correspond to the downward position of the pendulum, while odd values correspond to the upward position.Near each of the asymptotically stable critical points, the trajectories are clockwise spirals that represent a decaying oscillation about the equilibrium solution.

Example 4: Whirling (3 of 6)

The wavy horizontal portions of the trajectories for larger values of |y| represent whirling motions of the pendulum.A whirling motion cannot continue indefinitely, since eventually the angular velocity y is so much reduced by damping that the pendulum can no longer go over the top, and instead begins to oscillate about its downward position.

Example 4: Separatrix (4 of 6)

The trajectories that enter the saddle points separate the phase plane into regions. Such a trajectory is called a separatrix. Each region contains exactly one of the spiral points. The initial conditions on x = and y = d /dt determine the position of an initial point (x, y) in the phase plane. The subsequent motion of the pendulum is represented by the trajectory passing through the initial point as it spirals toward the asymptotically stable critical point in that region.

Example 4: Basins of Attraction (5 of 6)

The set of all initial points from which the trajectories approach an asymptotically stable critical point is called the basin of attraction, or the region of asymptotic stability, for that critical point. Each asymptotically stable critical point has its own basin of attraction, which is bounded by the separatrices through the neighboring unstable saddle points. The basin of attraction is shownin blue on the graph.

Example 4: Asymptotic Stability (6 of 6)

An important difference between nonlinear autonomous systems and the linear systems x' = Ax discussed in Section 9.1 is illustrated by the pendulum equations.Recall that x' = Ax has only the single critical point x = 0 if detA 0. Thus if the origin is asymptotically stable, then not only do the trajectories that start close to the origin approach the origin, but every trajectory approaches the origin. In this case the critical point x = 0 is globally asymptotically stable.This property is not true in general for nonlinear systems, and thus it is important to determine, or estimate, the basins of attraction for each asymptotically stable critical point.