Handbook of mathematics for engineers and scienteists part 82 docx

Phase trajectories of a system of differential equations near an equilibrium point of the node type... The phase trajectories in the vicinity of the equilibrium point have the same patte

By convention, for clearness and convenience of interpretation of the results, t will be used

to designate the independent variable and will be treated as time A solution x = x(t),

y = y(t) of system (12.6.1.25) plotted in the plane x, y (the phase plane) is called a (phase)

trajectory of the system

A solution to system (12.6.1.25) will be sought in the form

x = k1e λt, y = k

On substituting (12.6.1.26) into (12.6.1.25), one obtains the characteristic equation for the

exponent λ:



a11– λ a12

a21 a22– λ



 =0, or λ2– (a11+ a22)λ + a11a22– a12a21=0 (12.6.1.27)

The coefficients k1and k2are found as

k1= Ca12, k2= C(λ – a11), (12.6.1.28)

where C is an arbitrary constant To two different roots of the quadratic equation (12.6.1.27)

there correspond two pairs of coefficients (12.6.1.28)

Denote the discriminant of the quadratic equation (12.6.1.27) by

D = (a11– a22)2+4a12a21. (12.6.1.29) Three situations are possible

1◦ If D > 0, the roots of the characteristic equation (12.6.1.27) are real and distinct

(λ1≠λ2):

λ1 , 2= 12(a11+ a22) 12√

D The general solution of system (12.6.1.25) is expressed as

x = C1a12e λ1t + C2a12e λ2t,

y = C1(λ1– a11)e λ1t + C2(λ2– a11)e λ2t, (12.6.1.30)

where C1 and C2 are arbitrary constants For C1 = 0, C2 ≠ 0and C2 = 0, C1 ≠ 0, the

trajectories in the phase plane x, y are straight lines Four cases are possible here.

1.1 Two negative real roots, λ1<0and λ2<0 This corresponds to a11+ a22<0and

a11a22–a12a21>0 The equilibrium point is asymptotically stable and all trajectories starting

within a small neighborhood of the origin tend to the origin as t → ∞ To C1=0, C2 ≠ 0

and C2=0, C1≠ 0there correspond straight lines passing through the origin Figure 12.4 a depicts the arrangement of the phase trajectories near an equilibrium point called a stable node (or a sink) The direction of motion along the trajectories with increasing t is shown

by arrows

O

( )a

stable node

(sink)

O

unstable node (source)

λ > 0, λ > 01 2

Figure 12.4 Phase trajectories of a system of differential equations near an equilibrium point of the node type.

1.2 λ1 > 0 and λ2 > 0 This corresponds to a11+ a22 > 0and a11a22– a12a21 > 0 The phase trajectories in the vicinity of the equilibrium point have the same pattern as in the preceding case; however, the trajectories go in the opposite direction, away from the

equilibrium point; see Fig 12.4 b An equilibrium point of this type is called an unstable node (or a source).

1.3 λ1>0and λ2<0(or λ1<0and λ2>0) This corresponds to a11a22– a12a21<0

In this case, the equilibrium point is also unstable, since the trajectory (12.6.1.30) with

C2=0goes beyond a small neighborhood of the origin as t increases If C1C2≠ 0, then the

trajectories leave the neighborhood of the origin as t → –∞ and t → ∞ An equilibrium

point of this type is called a saddle (or a hyperbolic point); see Fig 12.5.

O

λ > 0, λ < 0

(λ < 0, λ > )0 saddle

1 1 2 2

Figure 12.5 Phase trajectories of a system of differential equations near an equilibrium point of the saddle

type.

1.4 λ1=0and λ2= a11+ a22≠ 0 This corresponds to a11a22– a12a21=0 The general solution of system (12.6.1.25) is expressed as

x = C1a12+ C2a12e(a11 +a22 )t,

y = –C1a11+ C2a22e(a11 +a22 )t, (12.6.1.31)

where C1 and C2 are arbitrary constants By eliminating time t from (12.6.1.31), one obtains a family of parallel lines defined by the equation a22x – a12y = a12(a11+ a22)C1

To C2 =0 in (12.6.1.31) there corresponds a one-parameter family of equilibrium points

that lie on the straight line a11x + a12y=0

(i) If λ2 <0, then the trajectories approach the equilibrium point lying as t→ ∞; see

Fig 12.6 The equilibrium point x = y = 0 is stable (or neutrally stable) — there is no asymptotic stability

y

λ = 0, λ < 0

a x+a y= 0 1

11 12

2

Figure 12.6 Phase trajectories of a system of differential equations near a set of equilibrium points located on

a straight line.

(ii) If λ2>0, the trajectories have the same pattern as in case (i), but they go, as t→ ∞,

in the opposite direction, away from the equilibrium point The point x = y =0is unstable

2◦ If D <0, the characteristic equation (12.6.1.27) has complex-conjugate roots:

λ1 , 2 = α iβ, α= 12(a11+ a22), β = 12

|D|, i2= –1

The general solution of system (12.6.1.25) has the form

x = e αt [C1cos(βt) + C2sin(βt)],

y = e αt [C1∗ cos(βt) + C2∗ sin(βt)], (12.6.1.32)

where C1and C2are arbitrary constants, and C1∗ and C2∗are defined by linear combinations

of C1and C2 The following cases are possible

2.1 For α <0, the trajectories in the phase plane are spirals asymptotically approaching

the origin of coordinates (the equilibrium point) as t → ∞; see Fig 12.7 a Therefore the

equilibrium point is asymptotically stable and is called a stable focus (also a stable spiral point or a spiral sink) A focus is characterized by the fact that the tangent to a trajectory

changes its direction all the way to the equilibrium point

( )a

stable focus

(spiral sink)

( )b

unstable focus (spiral source)

stable focus

(spiral sink)

Figure 12.7 Phase trajectories of a system of differential equations near an equilibrium point of the focus type.

2.2 For α >0, the phase trajectories are also spirals, but unlike the previous case they

spiral away from the origin as t → ∞; see Fig 12.7 b Therefore such an equilibrium point

is called an unstable focus (also an unstable spiral point or a spiral source).

2.3 At α =0, the phase trajectories are closed curves, containing the equilibrium point

inside (see Fig 12.8) Such an equilibrium point is called a center A center is a stable

equilibrium point Note that there is no asymptotic stability in this case

O

α = 0

center

Figure 12.8 Phase trajectories of a system of differential equations near an equilibrium point of the center

type.

3◦ If D = 0, the characteristic equation (12.6.1.27) has a double real root, λ1 = λ2 =

1

2(a11+ a22) The following cases are possible.

3.1 If λ1= λ2= λ <0, the general solution of system (12.6.1.25) has the form

x = a12(C1+ C2t )e λt,

y = [(λ – a11)C1+ C2+ C2(λ – a11)t]e λt, (12.6.1.33)

where C1and C2are arbitrary constants

Since there is a rapidly decaying factor, e λt, all trajectories tend to the equilibrium point

as t → ∞; see Fig 12.9 a To C2=0there corresponds a straight line in the phase plane x, y The equilibrium point is asymptotically stable and is called a stable node (a sink) Such a

node is in intermediate position between a node from Item 1.1 and a focus from Item 2.1

unstable node (source)

λ = λ < 0

stable node

(sink)

Figure 12.9 Phase trajectories of a system of differential equations near an equilibrium point of the node type

in the case of a double root, λ1= λ2

3.2 If λ1 = λ2 = λ > 0, the general solution of system (12.6.1.25) is determined by formulas (12.6.1.33) The phase trajectories are similar to those from Item 3.1, but they

go in the opposite direction, as t → ∞, rapidly away from the equilibrium point Such an

equilibrium point is called an unstable node (a source); see Fig 12.9 b.

3.3 If λ1= λ2=0, which corresponds to

a11+ a22=0 and a11a22– a12a21=0

simultaneously, the general solution of system (12.6.1.25) is obtained by substituting λ =0 into (12.6.1.33) and has the form

x = a12C1+ a12C2t,

y = C2– a11C1– a11C2t

For a12 ≠ 0all trajectories are parallel straight lines As t → ∞, the trajectories go away

from the origin The equilibrium point is unstable

For clearness, the classification results for equilibrium points of systems of two linear constant-coefficient differential equations (12.6.1.25) are summarized in Table 12.5

Remark For general definitions of a stable and an unstable equilibrium point, see Section 7.3.

TABLE 12.5 Classification of equilibrium points for systems of constant-coefficient equations (12.6.1.25); the symbols◦ and ∗ indicate stable and unstable equilibrium points, respectively, where not clearly stated

DiscriminantD,

formula (12.6.1.29)

Roots of quadratic equation (12.6.1.27),λ1 andλ2

Conditions for coefficients of differential equations (12.6.1.25)

Type of equilibrium points or shape of phase trajectories

D >0

λ1 < 0 ,λ2 < 0 ,λ1 ≠ 2

λ1 > 0 ,λ2 > 0 ,λ1 ≠ 2 roots have unlike signs

λ1 = 0 ,λ2 < 0

λ1 = 0 ,λ2 > 0

a11 +a22 < 0 ,a11a22 –a12a21 > 0

a11 +a22 > 0 ,a11a22 –a12a21 > 0

a11a22 –a12a21 < 0

a11 +a22 < 0 ,a11a22 –a12a21 = 0

a11 +a22 > 0 ,a11a22 –a12a21 = 0

stable node unstable node saddle∗ parallel lines◦ parallel lines∗

D <0 λ1,2=α

iβ, α >0

λ1 , 2 =αiβ, α <0

λ1 , 2 = iβ, imaginary roots

a11 +a22 < 0 , (a11 –a22 ) + 4a 12a21 < 0

a11 +a22 > 0 , (a11 –a22 ) + 4a 12a21 < 0

a11 +a22 = 0 ,a11a22 –a12a21 > 0

stable focus unstable focus center◦

D =0 λ λ11==λ λ22<>00

λ1 =λ2 = 0

a11 +a22 < 0 , (a11 –a22 ) + 4a 12a21 = 0

a11 +a22 > 0 , (a11 –a22 ) + 4a 12a21 = 0

a11 +a22 = 0 ,a11a22 –a12a21 = 0

stable node unstable node saddle∗ parallel lines∗

12.6.2 Systems of Linear Variable-Coefficient Equations

12.6.2-1 Homogeneous systems of linear first-order equations

1◦ In general, a homogeneous linear system of variable-coefficient first-order ordinary

differential equations has the form

y 

1= f11(x)y1+ f12(x)y2+· · · + f1n(x)y n,

y 

2= f21(x)y1+ f22(x)y2+· · · + f2n(x)y n,

y 

n = f n1 (x)y1+ f n2 (x)y2+· · · + f nn (x)y n,

(12.6.2.1)

where the prime denotes a derivative with respect to x It is assumed further on that the functions f ij (x) are continuous of an interval a≤x≤b (intervals are allowed with a = – ∞

or/and b = + ∞).

Any homogeneous linear system of the form (12.6.2.1) has the trivial particular solution

y1= y2=· · · = y n=0

Superposition principle for a homogeneous system: any linear combination of particular

solutions to system (12.6.2.1) is also a solution to this system

2◦ Let

yk = (y k1 , y k2 , , y kn)T, y km = y km (x); k , m =1, 2, , n (12.6.2.2)

be nontrivial particular solutions of the homogeneous system of equations (12.6.2.1)

So-lutions (12.6.2.2) are linearly independent if the Wronskian determinant is nonzero:

W (x)≡









y11(x) y12(x) · · · y1n(x)

y21(x) y22(x) · · · y2n(x)

. .

y n1 (x) y n2 (x) · · · y nn (x)







≠ 0. (12.6.2.3)

If condition (12.6.2.3) is satisfied, the general solution of the homogeneous system (12.6.2.1) is expressed as

where C1, C2, , C n are arbitrary constants The vector functions y1, y2, , y n in

(12.6.2.4) are called fundamental solutions of system (12.6.2.1).

3◦ Suppose condition (12.6.2.3) is met Then the Liouville formula

W (x) = W (x0) exp

 x

x0

n

s=1

f ss (t)



dt



holds

COROLLARY Particular solutions (12.6.2.2) are linearly independent on the interval

[a, b] if and only if there exists a point x0[a, b]such that the Wronskian determinant is

nonzero at x0: W (x0)≠ 0

4◦ Suppose a nontrivial particular solution of system (12.6.2.1),

y1= (u1, u2, , u n)T, u m = u m (x), m=1,2, , n,

is known Then the number of unknowns can be reduced by one To this end, one considers

the auxiliary homogeneous linear system of n –1equations

z 

k=

n



q=2



f kq (x) – u u k (x)

1(x) f1 (x)



z q, k=2, , n (12.6.2.5)

Let

zp = (z p , z p , , z pn)T, z mk = z mk (x); p=2, , n,

be a fundamental system of solutions to system (12.6.2.5) and let

Zp = (0, zp , z p , , z pn)T, F p (x) =

u1(x)

n



s=2

f1 (x)z ps (x)



dx, p=2, , n,

the vector Zphaving an additional component compared with zp Then the vector functions

yp = F p (x)y1+ zp (p =2, , n) together with y1form a fundamental system of solutions

to the initial homogeneous system of equations (12.6.2.1)

12.6.2-2 Nonhomogeneous systems of linear first-order equations

1◦ In general, a nonhomogeneous linear system of variable-coefficient first-order

differ-ential equations has the form

y 

1= f11(x)y1+ f12(x)y2+· · · + f1n(x)y n + g1(x),

y 

2= f21(x)y1+ f22(x)y2+· · · + f2n(x)y n + g2(x),

y 

n = f n1 (x)y1+ f n2 (x)y2+· · · + f nn (x)y n + g n (x).

(12.6.2.6)

Alternatively, the system can be written in the short vector-matrix notation as

y = f(x)y + g(x), with f(x) = f ij (x)

being the matrix of equation coefficients and g(x) = g1(x), g2(x),

, g n (x)T

being the vector function defining the nonhomogeneous part of the equations

EXISTENCE AND UNIQUENESS THEOREM Let the functions f ij (x) and g i (x)be

continu-ous on an interval a < x < b Then, for any set of values x ◦ , y ◦1, , y ◦ n , where a < x ◦ < b, there exists a unique solution y1= y1(x), , y n = y n (x)satisfying the initial conditions

y1(x ◦ ) = y ◦1, ., y n (x ◦ ) = y ◦ n,

and this solution is defined on the whole interval a < x < b.

2◦ Let

¯y = (¯y1,¯y2, , ¯y n)T, ¯y k= ¯y k (x); k=1, 2, , n,

be a particular solution to the nonhomogeneous system of equations (12.6.2.6) The general solution of this system is the sum of the general solution of the corresponding homogeneous

system (12.6.2.1), which corresponds to g k (x)≡ 0in (12.6.2.6), and any particular solution

of the nonhomogeneous system (12.6.2.6), or

where y1, y2, , y n are linearly independent solutions of the homogeneous system (12.6.2.1)

3◦ Given a fundamental system of solutions y km (x) (12.6.2.2) of the homogeneous system

(12.6.2.1), a particular solution of the nonhomogeneous system (12.6.2.6) is found as

¯y k=

n



m=1

y mk (x)



W m (x)

W (x) dx, k=1,2, , n,

where W m (x) is the determinant obtained by replacing the mth row in the Wronskian determinant (12.6.2.3) by the row of free terms, g1(x), g2(x), , g n (x), of equation

(12.6.2.6) The general solution of the nonhomogeneous system (12.6.2.6) is given by (12.6.2.7)

4◦ Superposition principle for a nonhomogeneous system A particular solution, y =¯y, of

the nonhomogeneous system of linear differential equations,

y = f(x)y +

m



k=1

gk (x),

is given by the sum

y =

m



k=1

yk,

where the yk are particular solutions of m (simpler) systems of equations

y k = f(x)y k+ gk (x), k=1, 2, , m, corresponding to individual nonhomogeneous terms of the original system

Remark For general definitions of a stable and. .. data-page="5">

TABLE 12.5 Classification of equilibrium points for systems of constant-coefficient equations (12.6.1.25); the symbols◦ and ∗ indicate stable and unstable equilibrium points,... continuous of an interval a≤x≤b (intervals are allowed with a = – ∞

or /and b = + ∞).

Any homogeneous linear system of the form (12.6.2.1) has the trivial particular