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Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone
The present investigation deals with global instability of a general -dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.
Recently, there has been a rapidly growing interest in investigating the instability conditions of differential systems. The number of papers dealing with instability problems is rather low compared with the huge quantity of papers in which the stability of the motion of differential systems is investigated. The first results on the instability of zero solution of differential systems were obtained in a general form by Lyapunov  and Chetaev .
Further investigation on the instability of solutions of systems was carried out to weaken the conditions of the Lyapunov and Chetaev theorems for special-form systems. Some results are presented, for example, in [3–10], but instability problems are analysed only locally. For example, in , a linear system of ordinary differential equations in the matrix form is considered, and conditions such that the corresponding forms (of the second and the third power) have fixed sign in some cone of the space are derived. To investigate this property another problem inverse to the known Lyapunov problem for the construction of Lyapunov functions is solved.
In the present paper, instability solutions of systems with quadratic right-hand sides is investigated in a cone dealing with a general -dimensional system with quadratic right-hand sides. We assume that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts.
Unlike the previous investigations, we prove the global instability of the zero solution in a given cone and the conditions for global instability are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. The main tool is the method of Chetaev and application of a suitable Chetaev-type function. A novelty in the proof of the main result (Theorem 3.1) is the utilization of a general third-order polynomial inequality of two variables to estimate the sign of the full derivative of an appropriate function along the trajectories of a given system in a cone.
In the sequel, the norms used for vectors and matrices are defined as for a vector and for any matrix . Here and throughout the paper, (or ) is the maximal (or minimal) eigenvalue of the corresponding symmetric and positive-semidefinite matrix (see, e.g., ).
In this paper, we consider the instability of the trivial solution of a nonlinear autonomous differential system with quadratic right-hand sides where coefficients and are constants. Without loss of generality, throughout this paper we assume As emphasized, for example, in [2, 10–12], system (1.3) can be written in a general vector-matrix form where is an constant square matrix, matrix is an rectangular matrix where the entries of the square matrices , are equal to zero except the row with entries , that is, and is a rectangular matrix such that where matrices , , that is, matrices are constant and symmetric. Representation (1.5) permits an investigation of differential systems with quadratic right-hand sides by methods of matrix analysis. Such approach was previously used, for example, in .
If matrix admits one simple positive eigenvalue, the system (1.5) can be transformed, using a suitable linear transformation of the dependent variables, to the same form (1.5) but with the matrix having the form where is an constant matrix, is the -dimensional zero vector and . With regard to this fact, we do not introduce new notations for the coefficients , in (1.5), assuming throughout the paper that in (1.5) has the form (1.10), preserving the old notations for entries of matrix . This means that we assume that , with for and , and , .
In this part we collect the necessary material-the definition of a cone, auxiliary Chetaev-type results on instability in a cone and, finally, a third degree polynomial inequality, which will be used to estimate the sign of the full derivative of a Chetaev-type function along the trajectories of system (1.5).
2.1. Instability of the Zero Solution of Systems of Differential Equations in a Cone
We consider an autonomous system of differential equations where satisfies a local Lipschitz condition and , that is, (2.1) admits the trivial solution. We will consider solutions of (2.1) determined by points where . The symbol denotes the solution of (2.1), satisfying initial condition .
Definition 2.1. The zero solution of (2.1) is called unstable if there exists such that, for arbitrary , there exists an with and such that .
Definition 2.2. A set is called a cone if for arbitrary and .
Definition 2.3. A cone is said to be a global cone of instability for (2.1) if for arbitrary and and .
Now, we prove results analogous to the classical Chetaev theorem (see, e.g., ) on instability in a form suitable for our analysis. As usual, if is a set, then denotes its boundary and its closure, that is, .
Theorem 2.5. Let , be a continuously differentiable function. Assume that the set is a cone. If the full derivative of along the trajectories of (2.1) is positive for every , that is, if then is a global cone of instability for the system (2.1).
Proof. Let be a positive number. We define a neighborhood of the origin
and a constant
Moreover, define a set
where is a positive number such that . Then, .
Let , then . We show that there exists a such that and .
Suppose to the contrary that this is not true and for all . Since , the function is increasing along the solutions of (2.1). Thus remains in . Due to the compactness of , there exists a positive value such that for Integrating this inequality over the interval , we get Then there exists a satisfying such that and, consequently, . This is contrary to our supposition. Since is arbitrary, we have that is, the zero solution is globally unstable, and is a global cone of instability.
Theorem 2.6. Let be a continuously differentiable function and let , be continuous functions such that . Assume that the set is a cone, and for any . If the full derivative (2.3) of along the trajectories of (2.1) is positive for every , that is, if for every , then is a global cone of instability for the system (2.1).
Proof. The proof is a modification of the proof of Theorem 2.5. Let be a positive number. We define a neighborhood of the origin by formula (2.4) and a constant
Moreover, define a set
where is a positive number such that . Then .
Let . Then . We show that there exists a such that and .
Suppose to the contrary that this is not true and for all . Since , the function is increasing along the solutions of (2.1). Due to the compactness of , there exists a positive value such that for Integrating this inequality over interval , we get Since , the inequality is an easy consequence of (2.15). Thus remains in . Apart from this, (2.15) also implies the existence of a satisfying such that . Consequently, . This is contrary to our supposition. Since is arbitrary, we have that is, the zero solution is globally unstable and is a global cone of instability.
Definition 2.7. A function satisfying all the properties indicated in Theorem 2.5 is called a Chetaev function for the system (2.1). A function satisfying all the properties indicated in Theorem 2.6 is called a Chetaev-type function for the system (2.1).
2.2. Auxiliary Inequality
Our results will be formulated in terms of global cones of instability. These will be derived using an auxiliary inequality valid in a given cone. Let and let be a positive number. We define a cone
Lemma 2.8. Let , , , , and be given constants such that , , , and . Assume, moreover, either or then for every .
Proof. We partition into two disjoint cones
and rewrite (2.23) as
We prove the validity of (2.23) in each of the two cones separately.
The case of the cone . Suppose that (2.20) holds. Estimating the left-hand side of (2.25), we get and (2.23) holds.
If inequalities (2.21) and (2.22) are valid, then, estimating the left-hand side of (2.25), we get and (2.23) holds again.
The case of the cone . Suppose that (2.20) hold, then, estimating the left-hand side of (2.25), we get and (2.23) holds.
If inequalities (2.21) and (2.22) are valid, then the estimation of (2.25) implies (we use (2.28)) Hence, (2.23) holds again.
3. Global Cone of Instability
In this part we derive a result on the instability of system (1.5) in a cone. In order to properly formulate the results, we have to define some auxiliary vectors and matrices (some definitions copy the previous ones used in Introduction, but with a dimension of rather than ). We denote Apart from this, we define symmetric matrices that is, Finally, we define an matrix where matrices , are defined as We consider a matrix equation where and are matrices. It is well-known (see, e.g., ) that, for a given positive definite symmetric matrix , (3.6) can be solved for a positive definite symmetric matrix if and only if the matrix is asymptotically stable.
Theorem 3.1 (Main result). Assume that the matrix is asymptotically stable, and is a positive number. Let be an positive definite symmetric matrix and be a related positive definite symmetric matrix solving equation (3.6). Assume that the matrix is positive definite,
and, in addition, one of the following conditions is valid:
either or a strong inequality holds in (3.7), and where Then the set is a global cone of instability for the system (1.5).
Proof. First we make auxiliary computations. For the reader's convenience, we recall that, for two matrices , , two vectors , , two vectors , and two “matrices" , , the multiplicative rule
holds. This rule can be modified easily for the case of arbitrary rectangular matrices under the condition that all the products are well defined.
We will rewrite system (1.5) in an equivalent form, suitable for further investigation. With this in mind, we define an matrix as where all the elements of the matrices , are equal to zero except the row, which equals , that is, Moreover, we define vectors , with components equal to zero except the th element, which equals , that is, and zero matrix .
It is easy to see that matrices and in (1.5) can be expressed as Now we are able to rewrite the system (1.5) under the above assumption regarding the representation of the matrix in the form (1.10) in an equivalent form Finally, since the equalities can be verified easily using (3.13), we have where The remaining part of the proof is based on Theorem 2.6 with a Chetaev-type function and with suitable functions and . Such functions we define as that is, We will verify the necessary properties. Obviously, , the set is a cone and for every .
The full derivative of (in the form (3.22)) along the trajectories of the system (1.5) (we use its transformed form (3.20)) equals Using formula (3.13), we get where We reduce these formulas using (3.21). Then, The derivative (3.26) turns into Finally, using (3.6), we get Let us find the conditions for the positivity of in the cone . We use (3.30). If , then and We set If in , then since is a positive definite matrix and If , then Now, we use Lemma 2.8 with , , , with coefficients , , , and defined by formula (3.32) and with .
Obviously because, due to (3.7), inequality holds. Moreover, if (3.8) holds, that is, if Further, if (3.9) holds, that is, if and (2.22) holds due to (4.10) and the condition . Thus the assumptions of Lemma 2.8 are true, the inequality (3.33) holds in the cone and, due to embedding (3.36), in the cone as well.
All the assumptions of Theorem 2.6 are fulfilled with regard to system (1.5) and the theorem is proved, because .
Remark 3.2. We will focus our attention to Lemma 2.8 about the positivity of a third-degree polynomial in two variables in the cone . We used it to estimate the derivative expressed by formula (3.30). Obviously, there are other possibilities of estimating its sign. Let us demonstrate one of them. Let us, for example, estimate the right-hand side of (3.31) in the cone using inequality (3.35), then and the positivity of will be guaranteed if We see that this approach produces only one set of inequalities for the positivity of , namely the case when (3.7) and (3.8) holds. Unfortunately, using such approach, we are not able to detect the second case (3.7) and (3.9) when is positive. This demonstrates the advantage of detailed estimates using the above third-degree polynomial in two variables.
4. Planar Case
Now we consider a particular case of the system (1.5) for . This means that, in accordance with (1.5) and (1.10), we consider a system where and . The solution of matrix equation (3.6) for , , and with , that is, gives with . The set defined by (3.12) where and reduces to Now, from Theorem 3.1, we will deduce sufficient conditions indicating being a global cone of instability for system (4.1). In our particular case, we have Now, we compute all necessary expressions used in Theorem 3.1. We have
Theorem 4.1 (Planar Case). Assume that , , , and . Let
and, in addition, one of the following conditions is valid:
either or strong inequality holds in (4.7), and where and are defined by (4.6). Then the set defined by (4.4) is a global cone of instability for the system (4.1).
It is easy to see that the choice , significantly simplifies all assumptions. Therefore we give such a particular case of Theorem 4.1.
Corollary 4.2 (Planar Case). Assume that , and . Let
and, in addition, one of the following conditions is valid:
either or strong inequality holds in (4.11), and where Then the set is a global cone of instability for the system (4.1).
This research was supported by Grants nos. P201/11/0768 and P201/10/1032 of Czech Grant Agency, and by the Council of Czech Government nos. MSM 0021630503, MSM 0021630519, and MSM 0021630529, and by Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication.
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