The Stability Criteria with Compound Matrices

Abstract

The bifurcation problem is one of the most important subjects in dynamical systems. Motivated by M. Li et al. who used compound matrices to judge the stability of matrices and the existence of Hopf bifurcations in continuous dynamical systems, we obtained some effective methods to judge the Schur stability of matrices on the base of the spectral property of compound matrices, which can be used to judge the asymptotical stability and the existence of Hopf bifurcations of discrete dynamical systems.

1. Introduction

The stability of matrices is intimately related to the stability of stationary solutions of various kinds in the theory and applications of dynamic systems [1]. Let 𝐴 be an 𝑛×𝑛 matrix, and let 𝜎(𝐴) be its spectrum. The stability modulus of 𝐴 is 𝑠(𝐴)=max{Re𝜆∶𝜆∈𝜎(𝐴)}, and 𝐴 is said to be stable if 𝑠(𝐴)<0. The asymptotic stability of the equilibrium point of a continuous dynamical system is closely related to the stability of the Jacobian matrix of the system at the equilibrium point; it is necessary that stability modulus is less than or equal to zero and it is sufficient that the Jacobian matrix is stable for the equilibrium point to be asymptotically stable. The stability of a matrix is related to the Routh-Hurwitz problem [2] on the number of zeros of a polynomial that have negative real parts. Li and Wang [3] have given a necessary and sufficient condition for the stability of an 𝑛×𝑛 matrix with real entries using the additive compound matrix. Accordingly, some criteria on the stability of the continuous dynamical systems are obtained.

The spectral radius of 𝐴 is 𝜌(𝐴)=max{|𝜆|∶𝜆∈𝜎(𝐴)}, and 𝐴 is said to be Schur stable if 𝜌(𝐴)<1. On the other hand, the asymptotic stability of a fixed point of a discrete dynamical system is closely related to the Schur stability of the linearized matrix of the system [4]; it is necessary that spectral radius of the linearized matrix is less than or equal to one and it is sufficient that the linearized matrix is Schur stable for the fixed point to be asymptotically stable. Corresponding to the Routh-Hurwitz criterion, the Jury criterion was derived by Jury in 1974. And Zheng et al. [5] obtained the extended Jury criterion on the coefficients of the characteristic polynomial of the matrix.

By the spectral property of the compound matrix, the compound matrix is an excellent tool to research the stability of matrices [6, 7]. In the present paper, using the additive compound matrix and the Lozinskii measure, some conclusions about the Schur stability of matrices and the stability of the discrete dynamical system are derived. Using the multiplicative compound matrix, we demonstrate how to judge the Schur stability of a matrix; at the same time we give the conditions that there exist no Hopf bifurcations in the discrete dynamical systems and the conditions for the fixed point to be asymptotically stable in the discrete dynamical systems.

We outline in the next section the preliminaries about the compound matrix and the Lozinskii measure and so on. In the Sections 3 and 4, the main results are demonstrated.

2. Preliminaries

Let 𝑀_𝑛(𝐓) be the linear space of 𝑛×𝑛 matrices with entries in 𝐓, where 𝐓 is the field of real numbers 𝐑 or complex numbers 𝐂. An 𝐴∈𝑀_𝑛(𝐓) will be identified with the linear operator on 𝐓^𝑛 that it represents. Let ∧ denote the exterior product in 𝐓^𝑛, and let 𝑘, 1≤𝑘≤𝑛, be an integer.

Definition 1 (see [6]). With respect to the canonical basis in the 𝑘th exterior product space ∧^𝑘𝐓^𝑛, the 𝑘th additive compound matrix 𝐴^[𝑘] of 𝐴 and the 𝑘th multiplicative compound matrix 𝐴^(𝑘) of 𝐴 are, respectively, linear operators on ∧^𝑘𝐓^𝑛 whose definitions on a decomposable element 𝑢₁∧⋯∧𝑢_𝑘 are 𝐴^[𝑘]𝑢₁∧⋯∧𝑢_𝑘=^𝑘_𝑗=1𝑢₁∧⋯∧𝐴𝑢_𝑗∧⋯∧𝑢_𝑘,∀𝑢₁,…,𝑢_𝑘∈𝐓^𝑛,𝐴^(𝑘)𝑢₁∧⋯∧𝑢_𝑘=𝐴𝑢₁∧⋯∧𝐴𝑢_𝑘,∀𝑢₁,…,𝑢_𝑘∈𝐓^𝑛. (1)
Definition over the whole ∧^𝑘𝐓^𝑛 is done by linear extension. The entries of 𝐴^[𝑘] and 𝐴^(𝑘) are linear relations of those of 𝐴.

Let 𝐴=(𝑎_𝑖𝑗),   and let 𝑁=𝐶^𝑘_𝑛. For any integer 𝑖=1,2,…,𝑁, let 𝑖=(𝑖₁,…,𝑖_𝑘) be the 𝑖th member in the lexicographic ordering of integer 𝑘-tuples such that 1≤𝑖₁<⋯<𝑖_𝑘≤𝑛. Then the following conclusions are concluded.

Proposition 2 (see [6]). We get the following: 𝐴^(𝑘)=𝑎_(𝑖)^(𝑗)_𝑁×𝑁. (2)
And let 𝑍=𝐴^[𝑘]=(𝑧_𝑖𝑗); then 𝑧_𝑖𝑗=𝑎_𝑖1𝑖1+⋯+𝑎_{𝑖𝑘𝑖𝑘}(𝑖)=𝑗,(),𝑖𝑓−1^𝑟+𝑠𝑎_{𝑗𝑟𝑖𝑠},𝑖𝑓𝑒𝑥𝑎𝑐𝑡𝑙𝑦𝑜𝑛𝑒𝑒𝑛𝑡𝑟𝑦𝑖_𝑠(𝑖)𝑗𝑜𝑓𝑑𝑜𝑒𝑠𝑛𝑜𝑡𝑜𝑐𝑐𝑢𝑟𝑖𝑛𝑎𝑛𝑑𝑗_𝑟(𝑖),(𝑖)𝑗𝑑𝑜𝑒𝑠𝑛𝑜𝑡𝑜𝑐𝑐𝑢𝑟𝑖𝑛0,𝑖𝑓𝑑𝑖𝑓𝑓𝑒𝑟𝑠𝑓𝑟𝑜𝑚𝑖𝑛𝑡𝑤𝑜𝑜𝑟𝑚𝑜𝑟𝑒𝑒𝑛𝑡𝑟𝑖𝑒𝑠. (3)

As special cases, we have 𝐴^[1]=𝐴 and 𝐴^[𝑛]=tr(𝐴). The spectral property is as follows.

Proposition 3 (see [6]). Let 𝜎(𝐴)={𝜆_𝑖∶𝑖=1,…,𝑛}: then(1)𝜎(𝐴^[𝑘])={𝜆_𝑖1+⋯+𝜆_𝑖𝑘∶1≤𝑖₁<⋯<𝑖_𝑘≤𝑛},(2)𝜎(𝐴^(𝑘))={𝜆_𝑖1⋯𝜆_𝑖𝑘∶1≤𝑖₁<⋯<𝑖_𝑘≤𝑛}.

Pertinent to the purpose of the present paper is the norm property of the compound matrix.

The singular values 𝜎₁,…,𝜎_𝑛 of an 𝑛×𝑛 matrix 𝐵 are the eigenvalues of the symmetric matrix √𝐵^∗𝐵, and let 𝜎₁≥⋯≥𝜎_𝑛≥0. (4)

Proposition 4 (see [7] (minimax-rule)). The following is established: 𝜎_𝑗=max_{𝑉⊂𝑅𝑚dim𝑉=𝑗}min_{‖𝑢‖𝑢∈𝑉=1}‖𝐵𝑢‖=min_{𝑊⊂𝑅𝑚dim𝑊=𝑚−𝑗+1}max_{‖𝑢‖𝑢∈𝑊=1}‖𝐵𝑢‖, (5)𝑗=1,…,𝑛, where ‖⋅‖ are the matrix norm induced from the Euclidean norm of 𝐑^𝑛.

Corollary 5 (see [7]). The following is established: 𝜎₁=‖𝐵‖.

Definition 6 (see [3]). Let |⋅| denote a vector norm in 𝐓^𝑛 and the operator norm it induces in 𝑀_𝑛(𝐓). For 𝐴∈𝑀_𝑛(𝐓), the Lozinskii measure 𝜇 on 𝑀_𝑛(𝐓) with respect to |⋅| is defined by 𝜇(𝐴)=lim_ℎ→0+|𝐼+ℎ𝐴|−1ℎ. (6)

The following are the stability criteria on matrices given by Li and Wang using the compound matrix and Lozinskii measure.

Proposition 7 (see [3]). We get the following: 𝑠(𝐴)<0⇔𝑠(𝐴^[2])<0 and (−1)^𝑛det(𝐴)>0.

Proposition 8 (see [3]). If (−1)^𝑛det(𝐴)>0, then 𝐴 is stable ⇔ there exists a Lozinskii measure 𝜇 on 𝑀_𝑁(𝐑) such that 𝜇(𝐴^[2])<0,𝑁=𝐶²_𝑛.

Definition 9 (see [3]). Let 𝛼↦𝐴(𝛼)∈𝑀_𝑛(𝐑) be a continuous function on (𝑎,𝑏). 𝛼₀∈(𝑎,𝑏) is called a Hopf bifurcation point of 𝐴(𝛼), if, when 𝛼<𝛼₀, 𝐴(𝛼) is stable. When 𝛼>𝛼₀, there exist a pair of plural eigenvalues Re𝜆(𝛼)±𝑖Im𝜆(𝛼) such that Re𝜆(𝛼)>0, and the other eigenvalues have nonzero real parts; here Im𝜆(𝛼)≠0.

Proposition 10 (see [3]). Consider the system 𝑥^′=𝑓(𝑥,𝛼),𝛼∈(𝑎,𝑏),𝑥∈𝐷,𝑓∈𝐶¹𝐑^𝑛×(𝑎,𝑏),𝐑^𝑛, (7) where 𝑓,𝜕𝑓/𝜕𝑥∈𝐶(𝐑^𝑛×(𝑎,𝑏),𝐑^𝑛) and 𝐷 is an open set in 𝐑^𝑛. If, for all 𝛼∈(𝑎,𝑏), 𝑓(𝑥,𝛼)=0 have solutions (𝑥(𝛼),𝛼), then 𝑥(𝛼) is called the equilibrium of the system (7). Let 𝐽(𝛼)=(𝜕𝑓/𝜕𝑥)(𝑥(𝛼),𝛼), and if 𝛼=𝛼₀ is the Hopf bifurcation point of 𝐽(𝛼), then the Hopf bifurcation of system (7) occurs when 𝛼=𝛼₀.

By the theorem, Li and Wang have given the methods to judge the existence of the Hopf bifurcation.

Proposition 11 (see [3]). If there exists some Lozinskii measure 𝜇 such that 𝜇𝐽^[2](𝛼)≤0,𝛼∈(𝑎,𝑏), (8) then the system (7) has no Hopf bifurcations from 𝑥(𝛼) in (𝑎,𝑏). Moreover, if (−1)^𝑛det(𝐽(𝛼))>0,𝛼∈(𝑎,𝑏), (9) then 𝑥(𝛼) is an asymptotically stable equilibrium of the system (7) for 𝛼∈(𝑎,𝑏).

3. The Schur Stability Criteria of Matrices Using the Additive Compound Matrix

Consider the 𝐶^𝑟(𝑟≥1) map: 𝑥⟼𝑔(𝑥),𝑥∈𝐑^𝑛. (10) If (10) has a fixed point 𝑥=𝑥, that is,  𝑥=𝑔(𝑥), then the linear map corresponding to (10) is 𝑦⟼𝐴𝑦,𝑦∈𝐑^𝑛, (11) where 𝐴≡𝐷𝑔(𝑥), the Jacobian matrix of 𝑔 at 𝑥.

Similar to the continuous dynamical system, the stability of a discrete dynamical system can be judged by the Schur stability of the matrix.

Lemma 12 (see [4]). If the matrix 𝐴 of the system (11) is Schur stable, then the fixed point 𝑥 of the system (11) is asymptotically stable.

It is generally well known that it is a sufficient condition for the Hopf bifurcation to occur in the system (10) that the matrix 𝐴 has a pair of conjugated eigenvalues whose spectral radius is equal to 1 (not equal to ±1), and the spectral radius of the rest of the eigenvalues is less than 1. In addition, it is a sufficient condition for the 𝑘-codimension Hopf bifurcation to occur in the system (10) that the matrix 𝐴 has 𝑘 pairs of conjugated of the eigenvalues whose spectral radius is equal to 1 (not equal to ±1), and the spectral radius of the rest eigenvalues is less than 1.

Thus, it is very necessary to investigate the following three questions:(1)the conditions for the spectral radius of all the eigenvalues of the matrix 𝐴 to be less than 1;(2)the conditions for the spectral radius of a pair of the eigenvalues of the matrix 𝐴 to be equal to 1 (not equal to ±1) and the spectral radius of the rest of the eigenvalues of the matrix 𝐴 to be less than 1;(3)the conditions for the spectral radius of 𝑘 pairs of the eigenvalues of the matrix 𝐴 to be equal to 1 (not equal to ±1) and the spectral radius of the rest of the eigenvalues of the matrix 𝐴 to be less than 1.

For the characteristic polynomial of the matrix 𝐴: 𝑓_𝑛(𝑧)=det𝑧𝐼_𝑛−𝐴=𝑧^𝑛+𝑎₁𝑧^𝑛−1+⋯+𝑎_𝑛, (12) we consider the operation 𝑧=𝜔+1,𝜔−1 (13) and (12) is changed into 𝑓_𝑛=𝜔^𝑛+𝑏₁𝜔^𝑛−1+⋯+𝑏_𝑛. (14)

By operation (13), the inner of the unit circle and the unit circle are mapped to left plane and the imaginary axis of the 𝜔 complex plane. And the questions about the eigenvalues of the matrices of the discrete dynamical systems are transformed into the questions about the eigenvalues of the matrices of the continuous dynamical systems. Therefore the following conclusions are obtained.

Theorem 13. If (−1)^𝑛det(𝐼+(2/det(𝐴−𝐼))(𝐴−𝐼)^∗)>0, then 𝐴 is Schur stable ⇔ there exists some Lozinskii measure 𝜇 such that 𝜇2𝐼+()()det𝐴−𝐼𝐴−𝐼^∗[2]<0. (15)

Proof. The eigenvalues of 𝐼+(2/det(𝐴−𝐼))(𝐴−𝐼)^∗ are 21+()⋅()det𝐴−𝐼det𝐴−𝐼2𝜆−1=1+=𝜆−1𝜆+1,𝜆−1 (16) where 𝜆 is the eigenvalue of the matrix 𝐴.
Therefore by (13), 𝜌(𝐴)2<1⟺𝑠𝐼+()()det𝐴−𝐼𝐴−𝐼^∗<0. (17) By Proposition 8, the conclusions are obtained.

Theorem 14. The spectral radius of a pair of conjugated eigenvalues of the matrix 𝐴 is equal to 1 (not equal to ±1) and the spectral radius of the rest eigenvalues is less than 1⇔ there exists a Lozinskii measure 𝜇 such that 𝜇(𝐼+(2/det(𝐴−𝐼))(𝐴−𝐼)^∗)^[3]<0, (−1)^𝑛det(𝐼+(2/det(𝐴−𝐼))(𝐴−𝐼)^∗)>0, 𝑠(𝐼+(2/det(𝐴−𝐼))(𝐴−𝐼)^∗)^[2]=0.

Proof. ⇒ It is obvious.
⇐ Let 2𝐼+()()det𝐴−𝐼𝐴−𝐼^∗=𝐴. (18) By 𝑠(𝐴)^[2]=0,  𝐴 has at most one eigenvalue that its real part is more than zero, and the rest of the eigenvalues have real parts less than zero. Without loss of generality, we assume that 𝜆_𝑛>0 and 𝜆_𝑛 is a real number. But signdet𝐴=(−1)^𝑛−1. (19) It contradicts (−1)^𝑛det𝐴>0. (20) And by 𝜇𝐴^[3]<0⟺𝑠𝐴^[3]<0, (21) it can be known that 𝐴 has only one pair of eigenvalues whose real parts are zero. Thus 𝐴 has only one pair of eigenvalues whose real parts are zero, and the real parts of the rest of the eigenvalues are less than zero. And the conclusions are established.

Consider the 𝐶^𝑟(𝑟≥1) map: 𝑥⟼𝑔(𝑥,𝛼),𝑥∈𝐑^𝑛,𝛼∈(𝑎,𝑏). (22) If (22) has the fixed point 𝑥=𝑥(𝛼), that is, 𝑥(𝛼)=𝑔(𝑥,𝛼), then the linear map corresponding to (22) is 𝑦⟼𝐴(𝛼)𝑦,𝑦∈𝐑^𝑛,𝛼∈(𝑎,𝑏), (23) where 𝐴≡𝐷𝑔(𝑥(𝛼),𝛼), the Jacobian matrix of 𝑔 at 𝑥(𝛼).

Definition 15. 𝛼₀∈(𝑎,𝑏) is called the Hopf bifurcation point if, when 𝛼<𝛼₀,  𝐴(𝛼) is Schur stable; when 𝛼>𝛼₀, there exist a pair of complex eigenvalues whose spectral radius is more than 1, and the rest of the eigenvalues are not 1 or −1.

Therefore the following theorem is obtained.

Theorem 16. If there exists some Lozinskii measure 𝜇 such that 𝜇(𝐼+(2/det(𝐽(𝛼)−𝐼))(𝐽(𝛼)−𝐼)^∗)^[2]≤0, 𝛼∈(𝑎,𝑏), then the system (22) has no Hopf bifurcation points about its fixed point 𝑥(𝛼). And if (−1)^𝑛det(𝐼+(2/det(𝐽(𝛼)−𝐼))(𝐽(𝛼)−𝐼)^∗)>0,𝛼∈(𝑎,𝑏), then 𝑥(𝛼) is an asymptotically stable fixed point of the system (22), where 𝑔,𝜕𝑔/𝜕𝑥∈𝐶(𝐑^𝑛×(𝑎,𝑏),𝐑^𝑛), 𝐽(𝛼)=(𝜕𝑔/𝜕𝑥)(𝑥(𝛼),𝛼), and 𝑥(𝛼) is the fixed point of the system (22).

4. The Schur Stability Criteria of Matrices Using the Multiplicative Compound Matrix

To research the Schur stability of the matrix and the stability of the discrete dynamical system, we find that the measure whose function is similar to that of the Lozinskii measure in the research of continuous dynamical systems is the norm of the matrix.

Let 𝑀_𝑛(𝐂) denote all the 𝑛×𝑛 matrices in the complex field.

Lemma 17. Let 𝐴∈𝑀_𝑛(𝐂); then 𝜌(𝐴)=inf{|𝐴|∶|⋅| is the matrix norm in 𝑀_𝑛(𝐂)}.

Thus we have the following corollary.

Corollary 18. Let 𝐴∈𝑀_𝑛(𝐂); then 𝜌(𝐴)<1⇔ there exists some matrix norm in 𝑀_𝑛(𝐂) such that |𝐴|<1.

By the spectral property of the multiplicative compound matrix, the Schur stability of the matrix can be judged by the multiplicative compound matrix.

Theorem 19. Let 𝐴∈𝑀_𝑛(𝐑); then 𝜌(𝐴)<1⇔𝜌(𝐴⁽²⁾)<1 and det(𝐼−𝐴²)>0.

Proof. ⇒ Because 𝜌(𝐴)<1, by Proposition 3, 𝜌(𝐴⁽²⁾)<1. It might be as well that the 𝑛 roots of 𝐴⁽²⁾ are 𝜈₁,…,𝜈_𝑠,𝜈_𝑠+1=𝜈₁,…,𝜈_2𝑠=𝜈_𝑠,𝜈_2𝑠+1,…,𝜈_𝑛, (24) where 𝜈_2𝑠+1,…,𝜈_𝑛 are real numbers. Then by 𝜌(𝐴)<1, we have 1−𝜈_𝑖1−𝜈_𝑖>0,𝑖=1,…,𝑠,1−𝜈_𝑗>0,𝑗=2𝑠+1,…,𝑛. (25) Thus det𝐼−𝐴²=^𝑠_𝑖=11−𝜈_𝑖1−𝜈_𝑖^𝑛_𝑗=2𝑠+11−𝜈_𝑗>0. (26)
⇐ By 𝜌(𝐴⁽²⁾)<1 and Proposition 3, 𝐴 has at most one eigenvalue whose spectral radius is more than 1. If 𝜌(𝐴)≥1, by 𝐴∈𝑀_𝑛(𝐑), it can be known that 𝐴 has just one real eigenvalue 𝜆₀ whose absolute value is more than or equal to 1. Let 𝑓(𝑥)=|𝑥𝐼−𝐴|=𝑥^𝑛−tr(𝐴)𝑥^𝑛−1+⋯+(−1)^𝑛|𝐴| (27) be the characteristic polynomial of 𝐴.
By lim_𝑥→+∞𝑓(𝑥)=+∞,lim_{𝑥→−∞}(−1)^𝑛𝑓(𝑥)=+∞, (28) it can be known that if 𝜆₀≥1, then det(𝐼−𝐴)≤0,(−1)^𝑛det(−𝐼−𝐴)>0. (29) Thus det𝐼−𝐴²<0, (30) a contradiction.
If 𝜆₀≤−1, then (−1)^𝑛det(−𝐼−𝐴)≤0,det(𝐼−𝐴)>0. (31) Thus det𝐼−𝐴²<0, (32) a contradiction.
Therefore the conclusions are established.

By the previous conclusions and corollaries, we have the following.

Corollary 20. Let 𝐴∈𝑀_𝑛(𝐑); then 𝜌(𝐴)<1⇔ there exists some matrix norm in 𝑀_𝑁(𝐑) such that |𝐴⁽²⁾|<1 and det(𝐼−𝐴²)>0, where 𝑁=𝐶²_𝑛.

By the property of the multiplicative compound matrix given in Section 2, the multiplicative compound matrix can be computed by the singular value of a matrix, and thus we have the following.

Corollary 21. Let𝐴∈𝑀_𝑛(𝐑); then 𝜌(𝐴)<1⇔𝜎₁𝜎₂<1 and det(𝐼−𝐴²)>0, 𝜎₁𝜎₂ are the singular values of 𝐴.

Corresponding to the stability of the fixed point and the Hopf bifurcation of the discrete dynamical systems, we have the following theorem.

Theorem 22. If there exists some matrix norm in 𝑀_𝑁(𝐑) such that |𝐽⁽²⁾(𝛼)|≤1, 𝛼∈(𝑎,𝑏), then the system (22) has no Hopf bifurcations about its fixed point 𝑥(𝛼). And if det(𝐼−𝐴²)>0, 𝛼∈(𝑎,𝑏), then 𝑥(𝛼) is an asymptotically stable fixed point of the system (22), where 𝑔,𝜕𝑔/𝜕𝑥∈𝐶(𝐑^𝑛×(𝑎,𝑏),𝐑^𝑛), 𝐽(𝛼)=(𝜕𝑔/𝜕𝑥)(𝑥(𝛼),𝛼), and 𝑥(𝛼) is the fixed point of the system (22).

Acknowledgments

This research is supported by the National Natural Science Foundation of China (no. 10871056) and the Fundamental Research Funds for the Central Universities (no. HIT. NSRIF. 2011001).