Open access peer-reviewed chapter

Characteristic Polynomials

Written By

Sándor Kovács, Szilvia György and Noémi Gyúró

Reviewed: August 30th, 2021 Published: February 25th, 2022

DOI: 10.5772/intechopen.100200

From the Edited Volume

Edited by Kamal Shah

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Abstract

In this chapter, we provide a short overview of the stability properties of polynomials and quasi-polynomials. They appear typically in stability investigations of equilibria of ordinary and retarded differential equations. In the case of ordinary differential equations we discuss the Hurwitz criterion, and its simplified version, the Lineard-Chippart criterion, furthermore the Mikhailov criterion and we show how one can prove the change of stability via the knowledge of the coefficients of the characteristic polynomial of the Jacobian of the given autonomous system. In the case of the retarded differential equation we use the Mikhailov criterion in order to estimate the length of the delay for which no stability switching occurs. These results are applied to the stability and Hopf bifurcation of an equilibrium solution of a system of ordinary differential equations as well as of retarded dynamical systems.

Keywords

• Hurwitz stability
• Schur stability
• Mikhailov criterion
• Hopf bifurcation

1. Introduction

As it is well-known, many systems of applied mathematics are modeled by retarded functional differential equations of type

ẋ=fxxτE1

(cf. [1]), where f:Ω×ΩRdis continuously differentiable and ΩRdis an open set. Here τ0represents the so-called delay or time lag. In order to have a solution in some interval 0r, r>0one has to know the solution on τ0, which means that one has to attach a continuous initial function ϕ:τ0Rdas an initial condition to the system (cf. [2]). Clearly, in the case of τ=0one has to deal with the initial value problem for ordinary differential equations.

In order to examine the stability of an equilibrium aΩof system (1), i.e. the equilibrium solution

â:RR,âtaE2

for which faa=0holds, one has to discuss the spectral properties of the linearized system

u̇=Au+BuτE3

where

A1faaRd×dandB2faaRd×d.E4

It may be supposed that system (3) has a solution of the form Rteλtswhere 0sRd, that is

λIdABeλτs=0.E5

This can only happen if and only if Δλτ=0holds, where

ΔzτdetzIdABezCE6

is called characteristic quasi-polynomial of the linear delay system (3).

The organization of the chapter is as follows. In the next section, we introduce and prove two criteria regarding stability of ΔΔ0, i.e. we give conditions for which the zeros of Δhave negative real parts. Concretely, we deal with the Hurwitz criterion and with its simplified version, the Lineard-Chippart criterion, and the Mikhailov criterion. Furthermore, we show how one can check the conditions of Hopf bifurcation via knowledge of the coefficients of the characteristic polynomial. In the section that follows we examine the case when the delay τis positive. We show how the Mikhailov criterion can be extended for quasi-polynomials and how the length of the delay can be estimated in order to have stability. Finally, we present a criterion for Hopf bifurcation.

2. The undelayed case: τ=0

If there is no delay present, i.e. τ=0holds then we have to deal with the characteristic polynomial

where the coefficients of χAare determined recursively by the Faddeev–LeVerrier algorithm (cf. [3, 4]) as follows

2.1 The stability of the characteristic polynomial χA

The asymptotic stability of (3) is determined by the stability of the matrix A, i.e. by the stability of its characteristic polynomial χA. We are now supplying some criteria for the stability of the characteristic polynomial χA. Under stability, we mean the so-called Hurwitz stability, i.e. the zeros of χAlie in the open left half of the complex plane. In this case χAis called also Hurwitz polynomial.

There is a very simple but very important necessary condition for χAto be Hurwitz (cf. [5, 6]).

Theorem 1.1 (Stodola). If the characteristic polynomial χAin (7) is stable then all of its coefficients are positive, i.e. ak>0holds where k0d1.

Proof:The real and complex zeros of χAmy be written as λk, resp. αl±iβl, where αland βlare both real. If the multiplicity of the real, resp. complex zeros are denoted by σk, resp. τl, where k1r, resp. l1s, then

k=1rσk+2l=1sτl=d,E9

and we can split χAinto linear, resp. quadratic factors according to the real, resp. complex zeros as follows

Thus, the stability of χAimplies the sign conditions

λk<0,αl<0k1rl1s.E11

This means that all coefficients of all factors in the product above are positive. By performing the multiplications one can see that the coefficients of χAare positive.

In the case of d=1and d=2this criterion is sufficient and necessary. Indeed, in case of d=1the characteristic polynomial has the form

χAz=z+a0zC,E12

furthermore in case of d=2we have

a1=TrA,a0=12TrA2TrA2=detAE13

(cf. (8)), thus

χAξ±=0ξ±=TrA±TrA24detA2E14

and χAis stable if and only if TrA<0and detA>0hold, because if

• TrA24detA>0, then we have ξ<0,

ξ+<0TrA24detA<TrA2detA>0;E15

• TrA24detA=0, then the zeros ξand ξ+are equal and real:

ξ=ξ+=TrA2<0TrA>0;E16

• TrA24detA<0, then there are only complex zeros:

ξ±=TrA2<0TrA>0.E17

Unfortunately the criterion is for d>2not sufficient. For example, the polynomial

pzz4+3z3+3z2+3z+3=z2+1z+1z+2zCE18

has positive coefficients, but two of its zero, namely ±iare not in the open left half-plane. In case of d=3there is a result which can be proved in several ways. Pontryagin (cf. [7]) proves it in a circumstantial way. He uses that the zeros of a polynomial are continuous functions of the coefficients (cf. [8, 9]). Our presentation is based on the results of Suter (cf. [10]).

Theorem 1.2In case of d=3the characteristic polynomial of Ahas the form

(cf. (8)) and χAis stable if and only if

hold

Proof:Using the Faddeev-LeVerrier-algorithm (cf. (8)) we have

χAz=z3+az2+bz+czC,E21

where

As a consequence of the fundamental theorem of algebra, we can split χAinto a linear and a quadratic factor

χAz=zαz2+βz+γ=z3+βαz2+γαβzαγzC.E23

In view of the above considerations for the first and the second-order polynomials we see that χAis stable if and only if α<0, β>0and γ>0hold. Thus, it is enough to show that the equivalence

α<0,β>0,γ>0a>0,c>0,abc>0.E24

holds. We prove this statement in two steps.

Step 1.We prove that the positivity of the coefficients a,b,centails

α<0,γ>0andsgnβ=sgnabc.E25

Indeed,

• from αγ=c>0it follows that α0, γ0. Hence the equivalence

α>0γ<0E26

holds. The case α>0,γ<0cannot happen, because γαβ=b>0would imply β<0, which is not possible due to βα=a>0.

α2+b>0E27
abc=βαγαβ+αγ=α2βαβ2+βγ=βα2αβ+γ==βα2+b.E28

Step 2.It remains to prove that the equivalence (24) holds.

• If χAis stable, i.e. α<0, β>0and γ>0hold, then a>0, b>0and clearly c>0, even by Step 2 sgnabc=sgnβ=1, i.e. abc>0.

• If inequalities a>0, c>0and abc>0hold, then b>0, hence by Step 2 we have α<0, γ>0and sgnβ=sgnabc=1, i.e. β>0which completes the proof.

Example 1.If the matrix AR3×3is antisymmetric, i.e. for suitable a,b,cR

A=0aba0cbc0E29

holds, then its characteristic polynomial has the form

χAz=z3+a2+b2+c2z=zz2+a2+b2+c2zC.E30

This means that χAand hence Ais unstable.

In order to formulate the necessary and sufficient stability condition for the polynomial χAwith arbitrary degree dN, we shall first fix our terminology. Let us define the Hurwitz matrix of the characteristic polynomial χAby

i.e.

Theorem 1.3(Routh-Hurwitz criterion). The characteristic polynomial χAin (7) is stable if and only if all leading principal minors

Δkdeth11h1khk1hkkk1dE33

of HχAare positive, i.e.

hold. □

For

• two-dimensional system we have

HχAa1a11a0=a101a0.E35

Thus, this criterion can be stated as

Δ1=a1>0,Δ2=a1a0>0,E36

or

a0>0,a1>0.E37

• third-dimensional system we have

HχAa2a001a100a2a0.E38

Thus, this criterion is

Δ1=a2>0,Δ2=a2a1a0>0,Δ3=a0Δ2>0,E39

or

a0>0,a2>0,a2a1>a0.E40

• fourth-dimensional system we have

HχAa3a1001a2a000a3a1001a2a0.E41

Thus, this criterion can be stated as Δ1=a3>0,

Δ2=a3a2a1>0,Δ3=a3a2a1a32a0a12>0,Δ4=a0Δ3>0E42

or

a0>0,a3>0,a3a2>a1,a1a2a3>a0a32+a12.E43

As the application of the above theorem, we mention the Orlando formula (cf. [11]) which establishes the useful relation between the Hurwitz determinants and the polynomial whose roots are sums of the roots of a given polynomial and which can be proved by mathematical induction (cf. [12]).

Theorem 1.4(Orlando-formula) If ξ1,,ξdare the roots of the characteristic polynomial χAthen the d1-th principal minor of the Hurwitz matrix can be expressed as

Δd1=1dd1/2i,j=1i<jdλi+λj.E44

In case of

• d=2this formula reduces to the well known Vieta formula in the quadratic equation

a1=Δ1=λ1+λ2;E45

• d=3the formula in (44) reduces to

a2a1a0=Δ2=λ1+λ2λ1+λ3λ2+λ3.E46

We remark (cf. [12]) if criterion (34) is satisfied then χAis stable which due to the form (10) has the consequence that all coefficients of χAare positive:

Clearly, if (47) holds then condition (34) is redundant: many of inequalities in (34) are unnecessary. For χAto be a Hurwitz stable, a necessary and sufficient condition can be established which requires about half amount of computations needed in the criterion of Routh-Hurwitz (cf. [12, 13]).

Theorem 1.5(Liénard-Chipart) The following statements are equivalent:

1. the characteristic polynomial χAin (7) is Hurwitz stable;

2. a0>0, a2>0, ; Δ1>0, Δ3>0, ;

3. a0>0, a2>0, ; Δ2>0, Δ4>0, ;

4. a0>0, a1>0, a3>0; Δ1>0, Δ3>0, ;

5. a0>0, a1>0, a3>0; Δ2>0, Δ4>0, .

Example 2.For α,βRthe matrix

A01000001000001000001αβ12321E48

has the characteristic polynomial

χAz1αβ+2z+3z2+2z3+z4zC.E49

It suffices to calculate

Δ4=detHχA=1αβΔ3=1αβdet220131αβ022=1αβ1241αβ4=41αβ1+αβ.E50

Thus, χAand hence Ais stable if and only if

1αβ>0and1+αβ>0,i.e.1<αβ<1E51

holds (cf. Figure 1).

There is a criterion of geometric character which is useful for the study of the stability of χA.

Definition 1.The curve

ΓAχA=(χAχA)C:ωRE52

is called Mikhailov hodograph or amplitude-phase curve (cf. [14]).

Some geometrical properties of the hodograph ΓAare in strong relationship to the stability of the characteristic polynomial χA. This polynomial has no zero on the imaginary axis if and only if the origin does not lie on the curve ΓA. In this case the function

φχAωargχA=logχAωRE53

is continuous in every point of the real line. Moreover, we deal with the change

ΔχAΔωφχAωE54

where ΔχAdenotes the change of argument of the vector φχAωin the complex plane as ωincreases from to +. Because

χA=χA¯,E55

therefore we have

Δω0φχAω=12ΔωφχAω.E56

This means that it is enough to know the behavior of the vector φχAωfor 0ω<. The next theorem which is known as the Mikhailov criterion of stability is based on the principle of argument. Because the form as it is in the next theorem is a special case of the one formulated in the next section we omit its proof now.

Theorem 1.6(Mikhailov). The polynomial χAis stable if and only if the following two conditions are fulfilled:

1. the curve ΓAdoes not cross the origin, i.e. the implication

χAz=0z0

is true, which means that χAhas no zeros on the imaginary axis;

2. the curve ΓAencircles the origin anticlockwise at an angle /2while ωchanges from 0to +, i.e.

Δω0argχA=2

holds.

Example 3.In case of d=2we have for the characteristic polynomial

χAza0+a1z+z2zC:E57
χA=a0ω2,χA=a1ω,E58

from which

sinargχA=χAχA2+p2=a1ωa0ω22+a12ω2sgna10ω+,E59

resp. as ω+

cosargχA=χAχA2+χA2=a0ω2a0ω22+a12ω21E60

follows. Thus,

Δω0+argχA=0a1<0a0<0,πa1<0a0>0,0a1>0a0<0,πa1>0a0>0E61

which means that χAis stable if and only if a0>0and a1>0hold.

Often what is to be checked is not the stability of the characteristic polynomial χAbut the question as to whether every zero of the polynomial lies in the interior of the unit circle around the origin of the complex plane. In this case χAis called Schur stable polynomial or simply Schur polynomial. This phenomenon plays a crucial role in the stability of discrete dynamical systems and in the asymptotic stability of periodic linear systems (cf. [15]). Regarding this problem there are two main treatments. The first way to investigate the Schur stability of χAis to introduce the Möbius-transformation

wz+1z11zC/zw+1w11wC/E62

which takes the interior of the unit circle of the complex plane zC:z<1into the interior of the left half-plane wC:w<0.Thus, if we want to know whether the polynomial χAis Schur stable we perform the transformation

ψAww1dχAw+1w11wC.E63

It is clear that ψAis also a polynomial of degree dand χAis Schur stable if and only if ψAis Hurwitz stable. It is not difficult to calculate (cf. [2]) that in case of

• d=2the polynomial χAis Schur stable if and only if

1+a1<a0<1;E64

• d=3the polynomial χAis Schur stable if and only if the inequalities

1+a1>a0+a2,3a1>3a0a2,1a1>a0a0+a2E65

hold.

The second way is the application of the so called Jury test which proof is based on the Rouché theorem (cf. [16]).

2.2 Hopf bifurcation

In what follows we shall examine the situation when system (1) with τ0exhibits Hopf bifurcation. In order to have this, we rewrite the version of (1) without delay in the parameter-dependent form

ẋ=fxpE66

where prepresents a parameter of the given system. Hopf bifurcation occurs if and only if for the eigenvalues μp±iνpof the Jacobi matrix Aof fat the critical value p

• the eigenvalue crossing condition holds:

μp=0,νp0,σA\±νp}iR=;E67

• the transversality condition μp0is fulfilled.

In the case of the two-dimensional system there is a result about the fulfillment of the above two conditions.

Lemma 1.Let IRbe an open interval and β,γ:IRsmooth functions. The roots of the characteristic polynomial

χAzz2+βz+γzCE68

fulfill the eigenvalue crossing condition and the transversality condition if and only if at the critical value p=pI

βp=0,γp>0andβp0E69

hold.

Proof:

Step 1.The polynomial χAhas purely imaginary zeros ±ωiwith ω0if and only if

z2+βz+γ=zωiz+ωi=z2+ω2zC.E70

Thus, at the critical value p=pIthe eigenvalue crossing condition holds exactly in case

βp=0andγp>0.E71

Step 2.Let denote by ρthe root of the equation

z2+βz+γ=0E72

which at ptakes the value ωi: ρp=ωi, and let us introduce the following function

Fzp:z2+βpz+γp.E73

Since

Fρpp=0and1Fρpp=2ωi+βp=2ωi0,E74

therefore we have

ρp=2Fωip1Fωip=βhz+γh2z+βhz=ωih=p=βpωi+γp2ωi+βpA+BiC+Di.E75

Using the well known calculation

furthermore the first and third part of (69) the formula

dρpdh=ρp=2ω2βp+γpβp4ω2+βp2=βp20E77

proves the lemma.

Because a matrix of order two can have no other eigenvalues besides the critical eigenvalues the crossing can happen only if for suitable r>0

β2h4γh<0hprp+rE78

holds.

We have to remark that there are two forms of Hopf bifurcation: the standard one and the so called non-standard. Under standard Hopf bifurcation we mean the phenomenon when the critical eigenvalues of the Jacobian matrix Across the imaginary axis from left to right and all other eigenvalues remain in the open left complex plane, whereas non-standard Hopf bifurcation means that the critical eigenvalues cross the imaginary axis from the right to the left and there is no restriction for the location of the other eigenvalues.

Theorem 1.7.If (78) holds then crossing can only happen

• From the left half-plane to the right, if the function βchanges at the critical values pits sign from positive to negative;

• From the right half-plane to the left, if the function βchanges at the critical values pits sign from negative to positive.

Proof:If condition (78) holds, then χAhas a pair of conjugate roots. From elementary mathematics, we know that

• if function βchanges its sign from positive to negative at the critical values p, then the sign of the real parts of the roots change their signs from the negative to positive;

• if function βchanges at the critical values pits sign from negative to positive, then the sign of the real parts of the roots change their signs from the positive to negative.

Example 4.Let be 0<a,bRand consider the activator-inhibitor system of Schnackenberg-type

ẋ=ax+x2y,ẏ=bx2yE79

(cf. [17]). System (79) has the unique equilibrium point

xy=a+bba+b2.E80

The Jacobian of (79) at xytakes the form

AJxy=2xy1x22xyx2E81

whose eigenvalues are the zeros of its characteristic polynomial

χAz=detzI2A=z2TrAz+detA=z2+12xy+x2z+x2=z2+ab+a+b3a+bz+a+b2z2+βz+γzC.E82

It is easy to see that if we choose bpas parameter by fixed athen for every p>0we have γp>0and

βp=1+3a+p2a+pa+pa+p3a+p2=2a+p32aa+p20.E83

This means that Hopf bifurcation occurs at the critical value pif and only if pis a positive real solution of the equation

κpa+pβh=ap+a+p3=0E84

extended in pto the whole real axis. For example,

• In case of a=1the polynomial κhas one real root:

κp=0p037i23+7i2.E85

Clearly, κ1=8>0, therefore κand so βassume on the positive half line positive values, which has a consequence that the characteristic polynomial and hence the equilibrium point xyis stable, since a second order characteristic polynomial is stable if and only if its coefficients have the same (positive) sign.

• in case of a=0.1the polynomial κhas three real roots:

κh=0hp11.18803p20.109149p30.778885.E86

Because

κ0=a+a3>0,κ0.5=0.184<0,κ1=aa2+3a+4>0,E87

the polynomial κand so βchanges its sign at p2from positive to negative, and at p3from negative to positive. This means in the light of the above that at the parameter value p=p2standard Hopf bifurcation occurs: the roots migrate from the left open half plane to the right, xyloses its stability; furthermore at the parameter value p=p3non-standard Hop bifurcation takes place, i.e. the roots migrate from the right half plane to the left and as a consequence xybecomes stable.

In the case of the three-dimensional system we post a result (cf. [18]), the proof of which is similar to the one in [19].

Lemma 2.Let be IRan open interval and α,β,γ:IRsmooth functions. The roots of the characteristic polynomial

χAzz3+αz2+βz+γzCE88

fulfill the crossing and the transversality conditions if and only if at the critical value p=pI

βp>0,αp0,γp=αpβpE89

furthermore

ddhαhβhγhh=p0E90

hold.

Proof:

Step 1.We show that the characteristic polynomial χAhas purely imaginary roots ±ωi(ω0), if αβγ=0and β>0hold. Indeed, if ξ,η,ζdenote the roots of the polynomial χA, then using Orlando formula we have

β=ξη+ξζ+ηζ,resp.αβγ=ξ+ηξ+ζη+ζ.E91

This means that

• if for some 0ωRthe equalities

χAωi=0=χAωiE92

hold, then one of the three zeros, like ζis real, furthermore

ξ=ωi=η,resp.β=ω2+ζωiζωi=ω2>0;E93

• χAhas exactly zeros with opposite sign but the same absolute value, if αβγ=0holds, furthermore χAhas a complex root, since if ξ=ηthen

β=ξ2+ξζξζ=ξ20,E94

which contradicts the fact that β>0.

It is clear that from conditions (89) it follows that the zeros of χAare

αand±βi=±γαi,E95

because

z+αzβiz+βiz+αz2+βz3+αz2+βz+αβz3+αz2+βz+γ.E96

Step 2.Let denote the roots of χAby ρwhich assumes at pthe value ωi: ρp=ωi, furthermore let define

Fzh:z3+αhz2+βhz+γh.E97

Because

Fρpp=0,and1Fωip=βp3ω2+2αpωi0,E98

it follows for the derivative of the implicit function ρat pthat

ρp=2Fωip1Fωip=αhz2+βhz+γh3z2+2αhz+βhz=ωih=p=αpω2βpωiγp3ω2+2αpωi+βp.E99

Using (89), resp. (90), furthermore βp=ω2we have

dρpdh=ρp=αpω2γpβp3ω22αpβpω2βp3ω22+4αp2ω2=αpβpγp2βp2αpβpβp2βp2+4αp2βp=2βpγpαpβpαpβp4βpβp+αp2==γαβp2βp+αp20.E100

As a consequence of the Stodola criterion the necessary condition for the stability of the polynomial χAis the positivity of its coefficients α,β,γ. In the case when its zeros cross the imaginary axis from the right to the left, one of the coefficients should be local negative.

Theorem 1.8.If conditions of Lemma 2., i.e. (89) and (90) hold, furthermore, for every ε>0, resp. arbitrary phpεp+εthe equalities

βh=α2handγh0ζhE101

are valid, where

ζh3αhβh2α2h9hI,E102

then the roots of χAcross the imaginary axis

• from left to right, if

3γhζh3<2αhE103

• from right to left, if

3γhζh3>2αhE104

hold.

Proof:Using the notations

aα3,bβ3,resp.Aa2b,B2a23ab+γE105

one can see that if A=0and 0BRhold then the zeros of the polynomial χAare as follows (cf. [17]):

ξ=B3a,η=B32a+3B234i,ζ=B32a+3B234i.E106

Thus, from B0and B32ait follows that χAhas a pair of complex conjugate roots. Because of condition γ0the third root could not be zero, furthermore this pair of complex conjugate zeros lies

• in the left half-plane if and only if B3<2a, i.e. (103) holds;

• in the right half-plane if and only if B3>2a, i.e. (104) holds.

Example 5.In [20] Liao, Zhou, and Tang proposed the following autonomous system of ordinary differential equations

ẋ=ayx,ẏ=dx+cyxz,ż=bz+xyE107

where a,b,c,dR. If abcdbelongs to the set

ab1d(abcca)(abc0)E108

then we get the Lorenz system (cf. [21]), the Chen system (cf. [22]) and the Lü system (cf. [23]). Yan showed (cf. [24]) that in the system (107) Hopf bifurcation may occur. In what follows we show that his calculations can be simplified as we know the coefficients of the characteristic polynomials of the Jacobian of system (107). If bc+d>0holds then system (107) has three equilibria:

E00,0,0,E±±bc+d±bc+dc+d.E109

The Jacobian of (107) takes the form

Jxyzaa0dzcxyxbxyzR3.E110

Hence the corresponding Jacobians are:

A0JE0aa0dc000b,E111

resp.

A±JE±aa0dc+dcbc+d±bc+d±bc+db.E112

The eigenvalues of A0are the roots of the characteristic polynomial

χA0ξdetξI3J0=ξ3+αξ2+βξ+γξC,E113

We remark that only a special parameter configuration was investigated in [24], namely a=c, d=2cc>0. On the other hand one can observe more due to Lemma 2:

1. it is easy to see from the characteristic polynomial that in case c=a+bHopf bifurcation may not occur because no matter what will be chosen as a bifurcation parameter, the coefficient of the first-order term of the polynomial vanishes, which has the consequence (cf. Lemma 2) that there can be a pair of complex conjugate zeros on the imaginary axis only in case if the constant term of the polynomial vanishes. This contradicts the crossing condition.

2. the parameter dcannot be considered as a bifurcation parameter, because as its value is changed Hopf bifurcation may not occur. This can be explained as follows. If a0then

γαβd=0d=ab+b2acbca

(in case of a=0the system would be two dimensional) and βd=b2, this contradicts the first condition in (89).

3. the parameter ccan be chosen as a bifurcation parameter only under some restrictions, because in the case of ab

and

βcaa+db2

will be positive only in case if aand a+dhave opposite signs as it is in case a=c, d=2cc>0proposed in [24]. If a=band a0then

γαβc=0c=ab+b2acbca

(in case of a=0the system would be two dimensional) and βc=b2which contradicts the first condition in (89). If a=cthen

γαβb=0d=0.

In this case we have

ddbγαβb=0,

It is easy to see the following: by fixing the parameters b,c, resp. d=2cc>0(cf. [24]) the parameter awill be chosen as bifurcation parameter then at value acHopf bifurcation takes place, because with substitution d=2cwe have

βa=cbbcccc2c=c2>0,αa=b0,E114

resp.

γαβa=abca+bcabbc+aca=c=bc2bc2=0E115

and

ddaγαβa=ddaabca+bcabbc+aca=c=bcabbc+aca+bcb+ca=c=bcc2bb+c=c2b20.E116

The eigenvalues of the matrix A±are the zeros of the characteristic polynomial

χA±ξdetξI3A±=ξ3+αξ2+βξ+γξCE117

where

If

a3a+da+b+d0,E119

then at

Hopf bifurcation takes place, because

βc=ba+d>0,αc=2aa+b+d3a+d,γαβc=0,E121

resp.

ddcγαβc=2abba+d=b3a+d0.E122

3. The delayed case: τ>0

When modeling and analyzing processes and behaviors which come from a natural environment, it often happens that we need a bit of distance in time to see the changes of the considered quantities (which are the variables of our model). For example, when we think about the epidemiological models, it is a well-founded thought that we need some time while susceptibles become infectious, and hence it is reasonable to assume that the migration of the individuals from the class of susceptibles into the infected is subject to delay.

Another expressive example is the modeling of the processes of the human body or the brain, like emotions: love, hate, etc. A bit of time has to pass for the brain to process the signals coming from various places, and only after this delay, the mood could change. These changes can be described and analyzed with delayed differential equations. One type of these systems is the so called Romeo and Juliet model, where the changes of Romeo’s and Juliet’s love and hate in time are described as a system of two linear ordinary differential equations. In this chapter we are going to consider this model with general coefficients and investigate the stability of the linear system.

We are going to consider the following linear system:

ẋ=α1x+A1xτ+α2y+A2yτ,ẏ=α3x+A3xτ+α4y+A4yτ,E123

where Ai, αiR, (i1,2,3,4), τ>0, with initial conditions Φ=Φ1Φ2in the Banach space

ΦCτ0R+2:Φ1θ=xθΦ2θ=yθ,E124

where Φiθ>0, (θτ0, i12). Straightforward calculation shows that the characteristic function of the above system (with regard to the trivial equilibrium point) takes the form:

Δzτz2α1+α4z+α1α4α2α3++e(A1+A4z+α1A4+α4A1α2A3α3A2)+e2A1A4A2A3zC.E125

In [25] the authors treat delay differential equations, which characteristic function for arbitrary zChas the form

Δzτ=z2+a1z+a0+b1z+b0e+ce2E126

where a0, a1, b0, b1, care arbitrary real constants. It can be seen that the characteristic functions (125) and (126) has the same form. In [25] the authors assume that c=0to simplify the analysis. Furthermore, they say that due to the continuous dependence of eigenvalues of the model parameters (cf. [26]) their results are valid for sufficiently small cparameters, too. Nevertheless, in the literature a lot of models and systems are investigated in which the coefficient of e2of the characteristic function is not equal to zero, and maybe not sufficiently small. Therefore, the aim of this section is to show that we can analyze the stability of the system in the case where c0, too.

In this section we assume that τ>0holds and investigate the qualitative behavior of the linearized system (3), more precisely we study the stability of characteristic function

Δzτ:pz+qze+rze2E127

where p, qand rare polynomials with real coefficients and degr<degq. Under stability of Δwe mean that the zeros of Δlie in the open left half of the complex plane. Using the Mikhailov criterion we give for special p, qand rfulfilling the above condition an estimate on the length of delay τfor which no stability switching occurs. Then for special parameters we compare our results with other methods. It follows then a delay independent stability analysis. Finally, a formula for Hopf bifurcation is calculated in terms of p, qand r. If we assume that the characteristic function has the form as in (126), then we can give conditions easily on the parameters a1,a0,b1,b0,cand an upper bound τ1such that with τ<τ1the system is asymptotically stable. In other words stability change may happen only for ττ1.

In what follows, the Mikhailov stability criterion will be proved, which is the implication of the argument principle (cf. [27, 28]). The treatment is based on [29].

Theorem 1.9(Mikhailov criterion). Consider the quasi-polynomial

MzQz+k=1pRkzesτkzC,E128

where the order of the polynomials Qand Rkis less than or equal to dN, and they are defined as

Qzqdzd++q0,Rkzrkdzd++rk0zCE129

where qi, rkiRfor i=1,,d, k=1,,p, qd>0and

maxk1pdegRk<d,E130

furthermore τk0for k=1,,p. If Mdefined by (128) has no zeros on the imaginary axis, then Mis stable if and only if

ΔΔω0+argM=2E131

holds where Δdenotes the change of argument of the vector Manticlockwise in the complex plane as ωincreases from 0to +.

Proof:In order to prove the theorem, we will apply the argument principle (cf. [28]) to Mon the Γ-contour (cf. Figure 2) where ΓC1C2denotes the positive oriented curve in the complex plane which consists of the interval ρρ(ρ>0) on the imaginary axis, i.e.

C1isC:sρρE132

and the semicircle C2of the radius ρin the right-hand half-plane:

C2ρeC:ϕπ2π2.E133

Since

maxk1pdegRk<degQ,E134

there is only a finite number of roots of Min the right-half plane.

On C2the characteristic equation can be written for every zCas follows:

Mz=qdρdcos+isin++q0+k=1prkd1ρd1cosd1ϕ+isind1ϕeρτke++rk0eρτke.E135

Now from the summation, we can write a typical term as

rkd1ρd1eid1ϕeρτke=rkd1ρd1eid1ϕeρτkcosϕiρτksinϕ=rkd1ρd1eρτkcosϕeid1ϕρτksinϕE136

Therefore,

Mz=qdρdcos+isin++q0+k=1peρτkcosϕrkd1ρd1cosd1ϕρτksinϕ+isind1ϕρτksinϕ+rkd2ρd2cosd2ϕρτksinϕ+isind2ϕρτksinϕ++rk0cosρτksinϕisinρτksinϕ.E137

Hence the argument or phase θof the vector Mzon C2may be written

tanθ=sinθcosθAB,E138

where

Asinθ=qdρdsin++q1ρsinϕk=1peρτkcosϕrkd1ρd1sind1ϕρτksinϕ++rk1ρsinϕρτksinϕrk0sinρτksinϕE139

and

Bcosθ=qdρdcos++q1ρcosϕ+q0+k=1peρτkcosϕrkd1ρd1cosd1ϕρτksinϕ++rk0cosρτksinϕ.E140

Dividing the numerator and denominator by ρdgives

tanθA1B1,E141

where

A1qdsin++q1ρd1sinϕ+k=1peρτkcosϕrkd1ρsind1ϕρτksinϕ++rk1ρd1sinϕρτksinϕrk0ρdsinρτksinϕE142

and

B1qdcos++q1ρd1cosϕ+q0ρd+k=1peρτkcosϕrkd1ρcosd1ϕρτksinϕ++rk0ρdcosρτksinϕ.E143

Now since

cosα1andsinα1αRE144

and since

cosα0απ2π2,E145

we have

tanθ=sincos=tanE146

or

θ=+,mE147

as ρ. Therefore, the change in argument of Mzon C2is given by

ΔC2argM=2+2+=.E148

Now from the argument principle we can write

ΔC1argM+ΔC2argM=2πN,E149

where Nis the total number of zeros of Minside Γ. Therefore

ΔC1argM=2πN.E150

If we reverse the direction of integration along C1and note the symmetry about the real axis, we have

Δω0+argM=122N=2πN.E151

in case of stability we have N=0. Hence, stability requires

Δω0+argM=2.E152

As a consequence, we have the following.

Lemma 3.Let p, qand rbe polynomials, with condition

degp>maxdegqdegr,E153

and assume that the quasi-polynomial in (127) has no roots on the imaginary axis. Then Δτis stable, i.e. all of its roots have negative real part if and only if

argΔτω=0ω=+=π2degp,E154

i.e. the argument of Δτincreases π/2degpas ωincreases from 0to +.

Theorem 1.10If for the delay parameter τin the characteristic function (126)

τ<a1b1b0+2c,a0+b0+c>0E155

hold, then the characteristic function, and hence the trivial equilibrium point of system (123) is asymptotically stable.

Proof:Substituting z=ω>0into (126), we get

Δτ=p+qeiωτ+re2iωτ.E156

Hence using the characteristic function (126), where

pzz2+a1z+a0,qzb1z+b0,rzcE157

we have for ω>0that

ΔRτω2+a0+b0cosωτ+b1ωsinωτ+ccos2ωτcsin2ωτ,E158

and also

ΔIτa1ω+b1ωcosωτb0sinωτ2csinωτcosωτ.E159

It could be seen that

ΔR0τ=a0+b0+c>0andΔI0τ=0,E160

furthermore

limω+ΔRτ=.E161

Therefore, we have to show that ΔIτ>0for each ω>0. If it holds, then

argΔτ=π,E162

and hence by using the Mikhailov criterion we have stability.

Substituting wωτinto (159) and multiplying the result by τ, we get

τΔIiwττ=a1w+b1wcoswτb0sinw2τcsinwcosw.E163

By using straightforward estimations we obtain that

τΔIiωττ>a1b1τb0+2cω,E164

and hence ΔIτ>0fulfills for ω>0, if the first condition of (155) is satisfied, i.e. if

τ<a1b1b0+2cE165

fulfills.

We show now a simple example in order to demonstrate the above theorem and what the conditions say. First of all, we are going to see that the conditions of the previous theorem are sufficient, but not necessary.

Example 6.Let us consider the following system of two linear delay differential equations.

ẋ=1+72x+xτ+4+72y12yτ,ẏ=x+xτ+3+72y.E166

The characteristic polynomial of (166) is

Δzτ=z2+2z+1+z1e+12e2zCτ0.E167

Since a0+b0+c=12>0, i.e. the second condition in Theorem 1.10 fulfills, we know from that theorem that if

τ<a1b1b0+2c=12,E168

then the trivial solution of (166) is asymptotically stable. As earlier mentioned, the conditions of Theorem 1.10 are sufficient, but not necessary, which can be easily seen, if we study the phase portrait of the system (166) with the following different values of the parameter τ: firstly with τ=0.48, then with τ=0.6and finally with τ=0.94. The Figures 3 and 4 represent the solutions of system (166) with different values of the parameter τ.

In Figure 3 (above) the parameter τis less than half, so the parameters fulfill the condition of Theorem 1.10, and the origin is asymptotically stable. In Figure 3 (bottom) the parameter value shows that the theorem does not give a necessary condition, because here the value of the parameter is bigger than half, but the origin is still asymptotically stable. But if we increase more the value of the parameter τ, the quasi-polynomial and hence the origin changes to unstable.

The previous example shows that it would be useful to give the largest bound in the theorem, because if we have a larger bound, then we can guarantee the stability of the quasi-polynomial for higher value of the parameter τ. In this sense, we can compare our result in Theorem 1.10 with another result in the literature. In this chapter, we compare the conditions of the theorem coming from [30, 31]. In [31] Stépán considered the system

ẋ=a11xa12y+b11xτ,ẏ=a21xa22yb22yτE169

where a11, a12, a21, a22, b11, b22>0and τ0. The characteristic function of system (169) is the quasi-polynomial

ΔSzτz2+a11+a22z+a11a22a12a21(b11+b22z+a22b11+a11b22)e+b11b22e2zCτ0.E170

In [30] Freedman and Rao worked with the system

ẋ=D1xD2y+B1xτ,ẏ=F1xF2y+E2yτE171

where B1, D1, D2, E2, F1, F2>0are constants and the characteristic function of (171) is the quasi-polynomial

ΔFRzτz2+D1+F2z+D1F2D2F1(B1+E2z+B1F2+D1E2)e+B1E2e2zCτ0.E172

Similarly to papers [30, 31] we gave an upper bound for τunder which the quasi-polynomial is Hurwitz stable.

Let us write these conditions for τusing the notations of Stépán. In [31] we can find the condition

τ<a11+a22b11b22b11b22+0.22a22+b22b11+0.22a11CS,E173

in [30] the condition

τ<D1+F2B1E22B1F2+D1E2+B1E2=a11+a22b11b222b11b22+a11b22+a22b11CFR.E174

Furthermore let us denote the right hand side of the condition in

Theorem 1.10by CGyK

τ<a1b1b0+2c=a11+a22b11+b222b11b22+a11b22+a22b11CGyK.E175

Since a11, a22, b11, b22>0, the numerators of CFRand CGyKare equal and

2b11b22+a11b22+a22b11=2b11b22+a11b22+a22b11<2b11b22+a11b22+a22b11,E176

because

a11b22+a22b11<2a11b22+a22b11E177

is true following from the positivity of these constants. This means that CGyK>CFR.

Repeatedly, following from the positivity of the constants a11, a22, b11and b22we get that the numerators of CGyKand CSare equal, but

b11b22+0.22a22+b22b11+0.22a11<2b11b22+a11b22+a22b11,E178

hence CGyK<CS. On one hand our result in Theorem 1.10 is applicable in general cases, because we have no additional constraints on the sign of the coefficients of the system (123), which means that in this sense our result is better. On the other hand we can increase the upper bound CSa little bit in the following way. In [31] Stépán used the estimation

sinx>0.22xx>0,E179

but actually this estimation is not sharp for positive x, cf. Figure 5.

If we find a tangent line of the sinefunction at a certain point x0, such that this line passes through the origin, i.e. the equation of this line is y=axwith a certain a<0, then we can get a better estimation than (179), namely

sinxaxx>0.E180

We can easily determine the constant a<0in the following way: the equation of the searched tangent line at x0is

y=fx0xx0+fx0=fx0x+fx0fx0x0,E181

where fis the sinefunction. We would like to find an x0such that

fx0fx0x0=sinx0cosx0x0=0E182

fulfills, which is true if and only if x0is a solution of the equation tanx=x. Let x0be the solution of this equation, then we find a better linear lower estimation for the function sineon the positive half-line

sinxcosx0xx>0,E183

i.e. acosx00.21724.

The proposition of the last theorem of this section is similar to one of Theorem 1.10. In the proof of Theorem 1.10 we used first-order Taylor polynomials to approximate the functions sineand cosineto obtain the estimation (164). But if we use higher-order polynomials, we can get a better result, i.e. a better estimation for τ, such that if τsatisfies the conditions, then the quasi-polynomial (126) is stable.

Theorem 1.11.If the coefficients of the characteristic function (125) fulfill the conditions

26105a0<59b0,421a1b1<0,32310531c<b0<32c,E184

and for the delay parameter τ

τ<κ+κ2ξC2E185

holds, where

κB2A,ξC4AE186

with

Ab02+124b0c+64c2,B6a1b0+32c4b1b0+2c,C45b12E187

then the quasi-polynomial (126) is Hurwitz-stable.

Proof:Firstly, let us make the same steps as in the proof of Theorem 1.10 and study the imaginary part of Δτ:

ΔIτ=a1ω+b1ωcosωτb0sinωτ2csinωτcosωτ.E188

To show that ΔIτ>0for each ω>0we apply the estimations

sinx<xx36+x5120,cosx>1x22x>0.E189

Then we have

ΔIτ>ωPτω,E190

with

Pτωωω4τ5120b0+32c+ω2τ26τb0+8c3b1+a1+b1τb0+2c.E191

If the conditions (184) are fulfilled, then the coefficient of ω4is positive in the polynomial Pτ, moreover if τsatisfies the condition (185) too, then the discriminant of Pτis negative. Hence with conditions (184) and (185) the inequality ΔIτ>0is valid for all ω>0. Similarly to the previous proof we get by applying the Mikhailov criterion that the quasi-polynomial is asymptotically stable.

In the following example, we are going to show that in some cases the result of Theorem 1.11 is better than the result in [31].

Example 7.Let us consider the following system of delay differential equations

ẋ=0.7xy+0.01xτ,ẏ=0.2x0.4y+0.07yτ.E192

The characteristic function of (192) is

Δzτz2+1.1z+22710000e225z+531000+e2710000zCτ0.E193

Let us see what condition gives [31] for the parameter τ. The condition

a11b11a22b22>a12a21E194

fulfills, hence if

τ<a11+a22b11b22b11b22+0.22a22+b22b11+0.22a1178.1,E195

then the quasi-polynomial (193) is asymptotically stable. Furthermore simple calculations show that following from Theorem 1.11, the system is asymptotically stable for all τ170.07, which is greater, than (195). In Figure 6 the phase portrait of the system (192) could be seen with some values of the parameter τ.

3.1 Stability investigation, independently of the delay

We are going to consider the general form of the characteristic function (126):

Δzτ=pz+qze+rze2zCE196

where degr<degq. We assume that if there is not any delay in the system, i.e. τ=0, then the trivial equilibrium point is asymptotically stable, which is equivalent to the assumption that the polynomial Δ0is Hurwitz stable. We know from [32] that with this assumption the system (and also its trivial equilibrium point) is delay-independently asymptotically stable if and only if for every τ>0the quasi-polynomial Δτhas no non-zero real root on the imaginary axis.

In this chapter we will add a condition to the polynomial p, qand rsuch that the mentioned property on the root of Δτfulfills. In the computations we will follow the idea of [33].

Firstly, let us multiply the equality Δτ=0by eiωτ, then we can see that the equivalence

Δτ=0eiωτΔτ=0E197

is valid. Let us introduce the notations

ppR+ipI,qqR+iqI,rrR+irI,ΔτΔRτ+iΔIτ.E198

With these notations the characteristic function (196) can be written at z=in the form

eiωτΔτ=eiωτp+q+reiωτ=pR+ipI(cosωτ+isinωτ+qR+iqI+rR+irI(cosωτisinωτ=ΔRτ+iΔIτ.E199

Let

xcosωτ2andysinωτ2,E200

then with straightforward calculations we can make the following transformations:

ΔRτ=cosωτpR+rR+qR+sinωτrIpI=cos2ωτ2sin2ωτ2pR+rR+cos2ωτ2+sin2ωτ2qR+2sinωτ2cosωτ2rIpIxxqR+pR+rR+yrIpI+yxqRpRrR+yrIpIxAxx+Ayy+yBxx+ByyAx+By.E201

Furthermore we can write the imaginary part in the same way, too:

ΔIτ=cosωτpI+rI+qI+sinωτrRpR=cos2ωτ2sin2ωτ2pI+rI+cos2ωτ2+sin2ωτ2qI+2sinωτ2cosωτ2rRpR=xxqI+pI+rI+yrRpR+yxqIpIrI+yrRpRxCxx+Cyy+yDxx+DyyCx+Dy.E202

Hence

eiωτΔτ=0AωτBωτCωτDωτxy=0.E203

Since the coefficients A, B, Cand Din the above matrix are expressed as the linear combination of xand y, we can write expressions eiωτΔRτand eiωτΔIτas follows

eiωτΔRτ=AxBxx2+AxBy+AyBxxy+AyByy2c0x2+c1xy+c2y2E204

and

eiωτΔIτ=CxDxx2+CxDy+CyDxxy+CyDyy2d0x2+d1xy+d2y2.E205

Then, by dividing the equalities eiωτΔRτ=0and eiωτΔIτ=0by y2, and introducing a new variable ux/y, we obtain

ΔRτ=y2c0u2+c1u+c2eiωτ,ΔIτ=y2d0u2+d1u+d2eiωτ.E206

Thus, since eiωτ/y20, the equation Δτ=0has no real non-zero root for any given τ>0if and only of the polynomials fand ghave no common real non-zero root, where fu, resp. gudenote the expressions for ΔR, resp. ΔI. This is equivalent to that resfg=detRfg0or if resfg=detRfg=0, then discrf<0and discrg<0, where the resultant of the polynomials fand gis defined as

Rfg=detc0c1c200c0c1c2d2d1d000d2d1d0=c0d0c2d22+c2d1c1d0c0d1c1d2,E207

and the discriminant of a polynomial Fuau2+bu+cis discrFb24ac.Hence we have proved the following statement.

Theorem 1.12.The characteristic function (196) has not a non-zero root on the imaginary axis if and only if the polynomial Δ0is Hurwitz stable, and resfg=detRfg0or if resfg=detRfg=0, then discrf<0and discrg<0, where

fu=c0u2+c1u+c2andgu=d0u2+d1u+d0,E208

where c0, c1, c2, d0, d1and d2are defined in (204) and (205).

Example 8.Let us consider again system (166), i.e.

ẋ=Ax+xτ+By12yτ,ẏ=x+xτ+CyE209

with

A1+7/2,B4+7/2,C3+7/2.E210

Straightforward calculations show that for all ωR

resfg=8ω40.5ω2ω20.250=0,E211
discrf=16ω2<0,discrg=16ω2ω20.520.E212

Thus, the discriminant of gis not negative, therefore the stability of system (209) changes at some value τof the delay (as we have seen in the previous example).

Figure 7 also shows the changing of the stability of system (209), with τ=0the origin is asymptotically stable, but with τ=2the origin changes to unstable. The solutions of system (209) with different values of the parameter τcan be seen on Figure 7. The stability of system (209) changes if the value of the parameter τincreases.

3.2 Hopf bifurcation

In this subsection we are going to see for which value of the delay τcould change the stability of the system (123). For this purpose we are going to give conditions on the coefficients p, qand rto obtain the value of the delay at which stability switch may occur. Let us assume that for ω>0the conditions of Theorem 1.12 do not fulfill, i.e. for the polynomials fand g(defined in (206)) the resultant is equal to 0and the discriminants of for gis nonnegative. Furthermore let us assume that τis a solution of Δiωτ=0. Let us denote by z3τthe root of the quasi-polynomial (196) that assumes z3τ=iωand the characteristic function Δτas a function of the parameter τby

Izτpz+qze+rze2zCτ>0.E213

Thus, we can determine the derivative of z3at τby the Implicit Function Theorem (cf. [34]):

z3τ=τIiωτzIiωτ=EzτDzτz=iω,τ=τE214

where Ezτzqz+2zrzeand Dzτpze+qzτqz+rz2zrze.To investigate and prove the occurrence of the Hopf bifurcation we have to see the sign of the real part of the above fraction. But since p, qand rare almost arbitrary polynomials, the fraction could be too complicated, that is why we introduce the following notation:

a+ibc+idEzτDzτz=iω,τ=τwitha+ibc+id=ac+bdc2+d2.E215

Since c2+d2>0, it is enough to consider the sign of ac+bd. We are going to use the notations introduced in (198) and along the lines of these we introduce the following notations, too:

ppR+ipI,qqR+iqI,rrR+irI.E216

(For sake of simplicity replacing ωby ωwe write PiωPfor PpIpRqIqRrIrR.)

Computing the exact value of a, b, cand dwe have:

a=2ωrRsinωτrIcosωτ12qI,b=2ωrRcosωτ+rIsinωτ+12qR,c=pR+rR2τrRcosωτ+pI+rI2τrIsinωτ+qRτqR,d=pI+rI2τrIcosωτ+pRrR+2τrRsinωτ+qIτqI.E217

Thus,

ac+bd=2ω12qRqIqIqR+rRqIrIqR+12qRpI+rIqIpR+rRcosωτ+rIqI+rRqR+12qRpRrR+qIpIrIsinωτ+cos2ωτrRpI+rIrIpR+rR+sin2ωτrIpRrRrRpIrI+sin2ωτrRpR+rIpI.E218

Using the elementary identities

zz¯zz¯,izzizz,E219

furthermore the Euler formula e±izcosz±isinz,we can simplify the enumerator of z3τas follows ac+bd=ωA,where

Aq¯q+2r¯r+2r¯q+q¯peiωτ+q¯reiωτ+2r¯pe2iωτ.E220

Therefore, Hopf bifurcation occurs if sgnA=±1holds.

4. Summary

The location of zeros of polynomials and quasi-polynomials as well is crucial in the point of view of the stability of ordinary and retarded differential equations. Namely, if the zeros of the characteristic polynomial of the linearized matrix lie in the open left half of the complex plane, then the constant solution of the particular equation is asymptotically stable. The main task of our work was to depict different methods which allow the investigation of the stability of characteristic (quasi-)polynomials, too. The second objective of this work was in the case of retarded differential equations to treat a method how to estimate the length of the delay for which no stability switching occurs. As an application, we showed a method to detect Hopf bifurcation in ordinary and retarded dynamical systems.

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Written By

Sándor Kovács, Szilvia György and Noémi Gyúró

Reviewed: August 30th, 2021 Published: February 25th, 2022