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Flexicurity CapitalismFoundations, Problems, and Perspectives$

Peter Flaschel and Alfred Greiner

Print publication date: 2012

Print ISBN-13: 9780199751587

Published to Oxford Scholarship Online: May 2012

DOI: 10.1093/acprof:oso/9780199751587.001.0001

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(p.199) Mathematical Appendix: Stability Theorems

(p.199) Mathematical Appendix: Stability Theorems

Source:
Flexicurity Capitalism
Publisher:
Oxford University Press

Local and Global Stability Issues

1. Local and Global Stability in a System of Differential Equations

Let x ˙ d x d t = f ( x ) , x R n be a system of n-dimensional differential equations that has an equilibrium point x° such that f(x°) = 0, where t is interpreted as “time”. The equilibrium point of this system is said to be locally asymptotically stable, if every trajectory starting sufficiently near the equilibrium point converges to it as t + . If stability is independent of the distance of the initial state from the equilibrium point, the equilibrium point is said to be globally asymptotically stable, or asymptotically stable in the large (see Gandolfo [1996, p. 333]).

2. Theorems on Stability of a System of Linear Differential Equations and on Local Stability of a System of Nonlinear Differential Equations

Theorem A.1 (Local stability / instability theorem; see Gandolfo [1996, pp. 360–362])

Let x ˙ i = f i ( x ) , x = [ x 1 , x 2 , , x n ] R n | i = 1 , 2 , , n ) be an n-dimensional system of differential equations that has an equilibrium point x o = [ x 1 o , x 2 o , , x n o such that f ( x o ) = 0 Suppose that the functions fi have continuous first-order partial derivatives, and consider the Jacobian matrix evaluated at the equilibrium point

J = [ f 11 f 12 f 1 n f 21 f 22 f 2 n f n 1 f n 2 f n n ]
where f i j = f i / x j ( i , j = 1 , 2 , , n ) are evaluated at the equilibrium point.

  1. (i) The equilibrium point of this system is locally asymptotically stable if all the roots of the characteristic equation | λ I J | = 0 have negative real parts.

  2. (ii) The equilibrium point of this system is unstable if at least one root of the characteristic equation | λ I J | = 0 has a positive real part.

  3. (iii) The stability of the equilibrium point cannot be determined from the properties of the Jacobian matrix if all the roots of the characteristic equation | λ I J | = 0 have nonpositive real parts but at least one root has zero real part.

Theorem A.2 (See Murata (1977, pp. 14–16)

Let A be an (n × n) matrix such that

A = [ a 11 a 12 a 1 n a 21 a 22 a 2 n a n 1 a n 2 a n n ]
  1. (i) (p.200) We can express the characteristic equation | λ I A | = 0 as

    | λ I A | = λ n + a 1 λ n 1 + a 2 λ n 2 + + a r λ n r + + a n 1 λ + a n = 0 ,
    (5.1)
    where
    a 1 = t r a c e A = i = 1 n a i i , a 2 = ( 1 ) 2 i j | a i i a j i a j i a j j | , , a r = ( 1 ) r i j k | a i i a j i a i k a j i a j j a j k a k i a k j a k k | ( r ) , , a n = ( 1 ) n det A .

  2. (ii) Let λ i ( i = 1 , 2 , , n ) be the roots of the characteristic equation (5.1). Then we have

    t r a c e J = i = 1 n a i i = i = 1 n λ i , det A = i = 1 n λ i

Routh-Hurwitz conditions

Theorem A.3 (Routh-Hurwitz conditions for stable roots in an n-dimensional system; cf. Murata [1977, p. 92] and Gandolfo [1996, pp. 221–222])1

All of the roots of the characteristic equation (5.1) have negative real parts if and only if the following set of inequalities is satisfied:

λ 2 + a 1 λ + a 2 = 0
Δ n = | a 1 a 3 a 5 a 7 0 1 a 2 a 4 a 6 0 0 a 1 a 3 a 5 0 0 1 a 2 a 4 0 0 0 a 1 a 3 0 0 0 0 0 a n | 0

(p.201) The following theorems A.4–A.6 are corollaries of theorem A.3.

Theorem A.4 (Routh-Hurwitz conditions for a two-dimensional system)

All of the roots of the characteristic equation

λ 2 + a 1 λ + a 2 = 0
have negative real parts if and only if the set of inequalities
a 1 0 , a 2 0
is satisfied.

(p.202) Remark on theorem A.5:

The inequality a 2 0 is always satisfied if the set of inequalities (5.2) is satisfied.

Theorem A.6 (Routh-Hurwitz conditions for a four-dimensional system)

All roots of the characteristic equation

λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0
have negative real parts if and only if the set of inequalities
a 1 0 , a 3 0 , a 1 a 2 a 3 0
(5.2)
is satisfied.

Theorem A.6 (Routh-Hurwitz conditions for a four-dimensional system)

All roots of the characteristic equation

λ 4 + a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4 = 0 ,
have negative real parts if and only if the set of inequalities
a 1 0 , a 3 0 , a 4 0 , Φ a 1 a 2 a 3 a 1 2 a 4 a 3 2 0 ,
(5.3)
is satisfied.

Remark on theorem A.6:

The inequality a 2 0 is always satisfied if the set of inequalities (5.3) is satisfied.

3. Theorems on Global Stability of a System of Nonlinear Differential Equations

Liapunov’s theorem and Olech's theorem

Theorem A.7 (Liapunov's theorem; cf. Gandolfo [1996, p. 410])

Let x ˙ = f ( x ) , x = [ x 1 , x 2 , , x n ] R n be an n-dimensional system of differential equations that has the unique equilibrium point x o = [ x 1 o , x 2 o , , x n o ] such that ( x o ) = 0. Suppose that there exists a scalar function V = V ( x x o ) with continuous first derivatives and with the following properties (1)–(5):

  1. 1. V = 0 ,

  2. 2. V = 0 if and only if xixo i= 0 for all 䁀 {1, 2, n},

  3. 3.

    v + a s x x o + ,

  4. 4.

    V ˙ = i = 1 n v ( x 1 x 1 o ) x ˙ i 0 ,

  5. 5.

    V ˙ = 0 i f a n d o n l y i f x i x i * = 0 f o r a l l i Є { 1 , 2 , , n } .

Then the equilibrium point x° of this system is globally asymptotically stable.

Remark on theorem A.7:

The function V = V ( x x o ) is called the “Liapunov function”.

Theorem A.8 (Olech’s theorem; cf. Olech [1963] and Gandolfo [1996, pp. 354–355])

Let x ˙ i = f i ( x 1 , x 2 ) ( i = 1 , 2 ) be a two-dimensional system of differential equations that has the unique equilibrium point x 1 , o x 2 o such that f i ( x 1 , o x 2 o ) = 0 ( i = 1 , 2 ) . Suppose that the functions fihave continuous first-order partial derivatives. Furthermore, suppose that the following properties (1)–(3) are satisfied:

  1. 1. f 1 x 1 + f 2 x 2 0 everywhere,

  2. 2. ( f 1 x 1 ) ( f 2 x 2 ) ( f 1 x 2 ) ( f 2 x 1 ) 0 everywhere

  3. 3. ( f 1 x 1 ) ( f 2 x 2 ) 0 everywhere, or alternatively, ( f 1 x 1 ) ( f 2 x 2 ) 0 everywhere

Then the equilibrium point of the above system is globally asymptotically stable. (p.203)

4. Theorems That Are Useful to Establish the Existence of Closed Orbits in a System of Nonlinear Differential Equations

Poincaré-Bendixson theorem and Hopf-bifurcation theorem

Theorem A.9 (Poincaré-Bendixson theorem; cf. Hirsch and Smale [1974, chapter 11])

Let x ˙ i = f i ( x 1 , x 2 ) ( i = 1 , 2 ) be a two-dimensional system of differential equations with the functions fi continuous. A nonempty compact limit set of the trajectory of this system, which contains no equilibrium point, is a closed orbit.

Theorem A.10 (Hopf bifurcation theorem for an n-dimensional system; cf. Guckenheimer and Holmes [1983, pp. 151–152], Lorenz [1993, p. 96], and Gandolfo [1996, p. 477])2

Let x ˙ = f ( x ; ε ) , x R n , ε R be an n-dimensional system of differential equations depending upon a parameter ε Suppose that the following conditions (1)–(3) are satisfied:

  1. (1) The system has a smooth curve of equilibria given by f ( x o ( ε ) ; ε ) = 0 ,

  2. (2) The characteristic equation | λ I D f ( x o ( ε 0 ) ; ε 0 ) | = 0 has a pair of purely imaginary roots λ ( ε 0 ) , λ ¯ ( ε 0 ) and no other roots with zero real parts, where D f ( x o ( ε 0 ) ; ( ε 0 ) is the Jacobian matrix of this system at ( x o ( ε 0 ) ; ( ε 0 ) with the parameter value ε

  3. (3) d { Re λ ( ε ) } d ε | ε = ε 0 0 , where Reλ (ε)is the real part of λ (ε)

Then, there exists a continuous function λ(γ) with ε(0)= ε0, and for all sufficiently small values of γ ≠ 0 there exists a continuous family of nonconstant periodic solution x(t, γ) for this dynamical system, which collapses to the equilibrium point γ → 0. The period of the cycle is close to 2 π / Im λ ( ε 0 ) , where Imγ(ε0) is the imaginary part of γ ε(〉0).

underline Remark on theorem A.10:

We can replace condition (3) in theorem A.10 by the following weaker condition (3a) (cf. Alexander and York [1978]):

(3a) For all ε which are near but not equal to ε0, no characteristic root has zero real part.

The following theorem by Liu (1994) provides a convenient criterion for the occurrence of the so called simple Hopf bifurcation in an n-dimensional system. The “simple” Hopf bifurcation is defined as the Hopf bifurcation in which all the characteristic roots except a pair of purely imaginary ones have negative real parts.

Theorem A.11 (Liu's theorem; see Liu [1994])

Consider the following characteristic equation with n ¯ ¯ 3 : :

λ n + a 1 λ n 1 + a 2 λ n 2 + + a n 1 λ + a n = 0

(p.204) This characteristic equation has a pair of purely imaginary roots and (n–2) roots with negative real parts if and only if the following set of conditions is satisfied:

Δ i 0 f o r a l l i Є { 1 , 2 , , n 2 } , Δ n 1 = 0 , a n 0
where Δ i ( i = 1 , 2 , , n 1 ) are Routh-Hurwitz terms defined as
Δ 1 = a 1 , Δ 2 = | a 1 a 3 1 a 2 | , Δ 3 = | a 1 a 3 a 5 1 a 2 a 4 0 a 1 a 3 | , , Δ n 1 = | a 1 a 3 a 5 a 7 0 0 1 a 2 a 4 a 6 0 0 0 a 1 a 3 a 5 0 0 0 1 a 2 a 4 0 0 0 0 a 1 a 3 0 0 0 0 0 0 a n 0 0 0 0 0 a n 1 0 0 0 0 0 a n 2 a n 0 0 0 0 a n 3 a n 1 |

The following theorems A.12–A.14 provide us with some convenient criteria for two-dimensional, three-dimensional, and four-dimensional Hopf bifurcations respectively. It is worth noting that these criteria provide us with useful information on the “nonsimple” as well as the “simple” Hopf bifurcations.

Theorem A.12

(p.205) The characteristic equation

λ 2 + a 1 λ + a 2 = 0
has a pair of purely imaginary roots if and only if the set of conditions
a 1 = 0 , a 2 0
is satisfied. In this case, we have the explicit solution λ = ± i a 2 , w h e r e i = 1.

Proof Obvious, because we have the solution λ = ( a 1 ± a 1 2 4 a 2 ) / 2 .

Theorem A.13 (cf. Asada (1995) and Asada and Semmler [1995])

The characteristic equation

λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0
has a pair of purely imaginary roots if and only if the set of conditions
a 2 0 , a 1 a 2 a 3 = 0
is satisfied. In this case, we have the explicit solution λ = a 1 , ± i a 2 , where = 1

Theorem A.14 (cf. Yoshida and Asada [2001], and Asada and Yoshida [2003])

Consider the characteristic equation

λ 4 + a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4 = 0
(5.4)
  1. (i) The characteristic equation (5.4) has a pair of purely imaginary roots and two roots with nonzero real parts if and only if either of the following set of conditions (A) or (B) is satisfied:

    1. (A) a 1 a 3 0 , a 4 0 , Φ a 1 a 2 a 3 a 1 2 a 4 a 3 2 = 0.

    2. (B) a 1 = a 3 = 0 , a 4 0.

  2. (ii) The characteristic equation (5.4) has a pair of purely imaginary roots and two roots with negative real parts if and only if the following set of conditions (C) is satisfied:

    1. (C) a 1 0 , a 3 0 , a 4 0. Φ a 1 a 2 a 3 a 1 2 a 4 a 3 2 = 0.

Remarks on theorem A.14:

  1. (1) The condition Φ = 0 is always satisfied if the set of conditions (B) is satisfied.

  2. (2) The inequality A 2 0 is always satisfied if the set of conditions (C) is satisfied.

  3. (3) We can derive theorem A.14 (ii) from theorem A.11 as a special case with n=4, although we cannot derive theorem A.14 (i) from theorem A.11.

Notes:

(1.) See also Gantmacher (1954) for many details that can be associated with these conditions and Brock and Malliaris (1989) for a compact representation of them.

(2.) See also Strogatz (1994) and Wiggins (1990) in this regard.