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.
- Hurwitz stability
- Schur stability
- Mikhailov criterion
- Hopf bifurcation
As it is well-known, many systems of applied mathematics are modeled by retarded functional differential equations of type
(cf. ), where is continuously differentiable and is an open set. Here represents the so-called delay or time lag. In order to have a solution in some interval , one has to know the solution on , which means that one has to attach a continuous initial function as an initial condition to the system (cf. ). Clearly, in the case of one has to deal with the initial value problem for ordinary differential equations.
In order to examine the stability of an equilibrium of system (1), i.e. the equilibrium solution
for which holds, one has to discuss the spectral properties of the linearized system
It may be supposed that system (3) has a solution of the form where , that is
This can only happen if and only if holds, where
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 , i.e. we give conditions for which the zeros of have 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 is 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:
If there is no delay present, i.e. holds then we have to deal with the characteristic polynomial
2.1 The stability of the characteristic polynomial
The asymptotic stability of (3) is determined by the stability of the matrix , i.e. by the stability of its characteristic polynomial . We are now supplying some criteria for the stability of the characteristic polynomial . Under stability, we mean the so-called Hurwitz stability, i.e. the zeros of lie in the open left half of the complex plane. In this case is called also Hurwitz polynomial.
Theorem 1.1 (Stodola). If the characteristic polynomial in (7) is stable then all of its coefficients are positive, i.e. holds where .
and we can split into linear, resp. quadratic factors according to the real, resp. complex zeros as follows
Thus, the stability of implies the sign conditions
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 are positive.
In the case of and this criterion is sufficient and necessary. Indeed, in case of the characteristic polynomial has the form
furthermore in case of we have
(cf. (8)), thus
and is stable if and only if and hold, because if
, then we have ,E15
, then the zeros and are equal and real:E16
, then there are only complex zeros:
Unfortunately the criterion is for not sufficient. For example, the polynomial
has positive coefficients, but two of its zero, namely are not in the open left half-plane. In case of there is a result which can be proved in several ways. Pontryagin (cf. ) 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. ).
(cf. (8)) and is stable if and only if
As a consequence of the fundamental theorem of algebra, we can split into a linear and a quadratic factor
In view of the above considerations for the first and the second-order polynomials we see that is stable if and only if , and hold. Thus, it is enough to show that the equivalence
holds. We prove this statement in two steps.
from it follows that , . Hence the equivalence
holds. The case cannot happen, because would imply , which is not possible due to .
If is stable, i.e. , and hold, then , and clearly , even by Step 2 , i.e. .
If inequalities , and hold, then , hence by Step 2 we have , and , i.e. which completes the proof.
holds, then its characteristic polynomial has the form
This means that and hence is unstable.
In order to formulate the necessary and sufficient stability condition for the polynomial with arbitrary degree , we shall first fix our terminology. Let us define the Hurwitz matrix of the characteristic polynomial by
where and if .
of are positive, i.e.
two-dimensional system we have
Thus, this criterion can be stated as
third-dimensional system we have
Thus, this criterion is
fourth-dimensional system we have
Thus, this criterion can be stated as ,
As the application of the above theorem, we mention the Orlando formula (cf. ) 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. ).
In case of
this formula reduces to the well known Vieta formula in the quadratic equation
the formula in (44) reduces to
Clearly, if (47) holds then condition (34) is redundant: many of inequalities in (34) are unnecessary. For to 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]).
the characteristic polynomial in (7) is Hurwitz stable;
, , ; , , ;
, , ; , , ;
, , ; , , ;
, , ; , , .
has the characteristic polynomial
It suffices to calculate
Thus, and hence is stable if and only if
holds (cf. Figure 1).
There is a criterion of geometric character which is useful for the study of the stability of .
is called Mikhailov hodograph or amplitude-phase curve (cf. ).
Some geometrical properties of the hodograph are in strong relationship to the stability of the characteristic polynomial . This polynomial has no zero on the imaginary axis if and only if the origin does not lie on the curve . In this case the function
is continuous in every point of the real line. Moreover, we deal with the change
where denotes the change of argument of the vector in the complex plane as increases from to . Because
therefore we have
This means that it is enough to know the behavior of the vector for . 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.
the curve does not cross the origin, i.e. the implication
is true, which means that has no zeros on the imaginary axis;
the curve encircles the origin anticlockwise at an angle while changes from to , i.e.
which means that is stable if and only if and hold.
Often what is to be checked is not the stability of the characteristic polynomial but 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 is 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. ). Regarding this problem there are two main treatments. The first way to investigate the Schur stability of is to introduce the Möbius-transformation
which takes the interior of the unit circle of the complex plane into the interior of the left half-plane Thus, if we want to know whether the polynomial is Schur stable we perform the transformation
It is clear that is also a polynomial of degree and is Schur stable if and only if is Hurwitz stable. It is not difficult to calculate (cf. ) that in case of
the polynomial is Schur stable if and only if
the polynomial is Schur stable if and only if the inequalities
The second way is the application of the so called Jury test which proof is based on the Rouché theorem (cf. ).
2.2 Hopf bifurcation
where represents a parameter of the given system. Hopf bifurcation occurs if and only if for the eigenvalues of the Jacobi matrix of at the critical value
the eigenvalue crossing condition holds:
the transversality condition is fulfilled.
In the case of the two-dimensional system there is a result about the fulfillment of the above two conditions.
fulfill the eigenvalue crossing condition and the transversality condition if and only if at the critical value
Thus, at the critical value the eigenvalue crossing condition holds exactly in case
which at takes the value : , and let us introduce the following function
therefore we have
Using the well known calculation
furthermore the first and third part of (69) the formula
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
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 cross 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.
From the left half-plane to the right, if the function changes at the critical values its sign from positive to negative;
From the right half-plane to the left, if the function changes at the critical values its sign from negative to positive.
if function changes its sign from positive to negative at the critical values , then the sign of the real parts of the roots change their signs from the negative to positive;
if function changes at the critical values its sign from negative to positive, then the sign of the real parts of the roots change their signs from the positive to negative.
The Jacobian of (79) at takes the form
whose eigenvalues are the zeros of its characteristic polynomial
It is easy to see that if we choose as parameter by fixed then for every we have and
This means that Hopf bifurcation occurs at the critical value if and only if is a positive real solution of the equation
extended in to the whole real axis. For example,
In case of the polynomial has one real root:
Clearly, , therefore and so assume on the positive half line positive values, which has a consequence that the characteristic polynomial and hence the equilibrium point is stable, since a second order characteristic polynomial is stable if and only if its coefficients have the same (positive) sign.
in case of the polynomial has three real roots:
the polynomial and so changes its sign at from positive to negative, and at from negative to positive. This means in the light of the above that at the parameter value standard Hopf bifurcation occurs: the roots migrate from the left open half plane to the right, loses its stability; furthermore at the parameter value non-standard Hop bifurcation takes place, i.e. the roots migrate from the right half plane to the left and as a consequence becomes stable.
fulfill the crossing and the transversality conditions if and only if at the critical value
This means that
if for some the equalities
hold, then one of the three zeros, like is real, furthermore
has exactly zeros with opposite sign but the same absolute value, if holds, furthermore has a complex root, since if then
which contradicts the fact that .
It is clear that from conditions (89) it follows that the zeros of are
it follows for the derivative of the implicit function at that
As a consequence of the Stodola criterion the necessary condition for the stability of the polynomial is the positivity of its coefficients . In the case when its zeros cross the imaginary axis from the right to the left, one of the coefficients should be local negative.
are valid, where
then the roots of cross the imaginary axis
from left to right, if
from right to left, if
one can see that if and hold then the zeros of the polynomial are as follows (cf. ):
Thus, from and it follows that has a pair of complex conjugate roots. Because of condition the third root could not be zero, furthermore this pair of complex conjugate zeros lies
in the left half-plane if and only if , i.e. (103) holds;
in the right half-plane if and only if , i.e. (104) holds.
where . If belongs to the set
then we get the Lorenz system (cf. ), the Chen system (cf. ) and the Lü system (cf. ). Yan showed (cf. ) 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 holds then system (107) has three equilibria:
The Jacobian of (107) takes the form
Hence the corresponding Jacobians are:
The eigenvalues of are the roots of the characteristic polynomial
where , and .
We remark that only a special parameter configuration was investigated in , namely , . On the other hand one can observe more due to Lemma 2:
it is easy to see from the characteristic polynomial that in case Hopf 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.
the parameter cannot be considered as a bifurcation parameter, because as its value is changed Hopf bifurcation may not occur. This can be explained as follows. If then
(in case of the system would be two dimensional) and , this contradicts the first condition in (89).
the parameter can be chosen as a bifurcation parameter only under some restrictions, because in the case of
will be positive only in case if and have opposite signs as it is in case , proposed in . If and then
(in case of the system would be two dimensional) and which contradicts the first condition in (89). If then
In this case we have
which contradicts the transversality condition.
It is easy to see the following: by fixing the parameters , resp. (cf. ) the parameter will be chosen as bifurcation parameter then at value Hopf bifurcation takes place, because with substitution we have
The eigenvalues of the matrix are the zeros of the characteristic polynomial
Hopf bifurcation takes place, because
3. The delayed case:
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:
where , , (), , with initial conditions in the Banach space
where , (, ). Straightforward calculation shows that the characteristic function of the above system (with regard to the trivial equilibrium point) takes the form:
In  the authors treat delay differential equations, which characteristic function for arbitrary has the form
where , , , , are arbitrary real constants. It can be seen that the characteristic functions (125) and (126) has the same form. In  the authors assume that to simplify the analysis. Furthermore, they say that due to the continuous dependence of eigenvalues of the model parameters (cf. ) their results are valid for sufficiently small parameters, too. Nevertheless, in the literature a lot of models and systems are investigated in which the coefficient of of 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 , too.
In this section we assume that holds and investigate the qualitative behavior of the linearized system (3), more precisely we study the stability of characteristic function
where , and are polynomials with real coefficients and . Under stability of we mean that the zeros of lie in the open left half of the complex plane. Using the Mikhailov criterion we give for special , and fulfilling the above condition an estimate on the length of delay for 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 , and . If we assume that the characteristic function has the form as in (126), then we can give conditions easily on the parameters and an upper bound such that with the system is asymptotically stable. In other words stability change may happen only for .
where the order of the polynomials and is less than or equal to , and they are defined as
where , for , , and
furthermore for . If defined by (128) has no zeros on the imaginary axis, then is stable if and only if
holds where denotes the change of argument of the vector anticlockwise in the complex plane as increases from to .
and the semicircle of the radius in the right-hand half-plane:
there is only a finite number of roots of in the right-half plane.
On the characteristic equation can be written for every as follows:
Now from the summation, we can write a typical term as
Hence the argument or phase of the vector on may be written
Dividing the numerator and denominator by gives
as . Therefore, the change in argument of on is given by
Now from the argument principle we can write
where is the total number of zeros of inside . Therefore
If we reverse the direction of integration along and note the symmetry about the real axis, we have
in case of stability we have . Hence, stability requires
As a consequence, we have the following.
and assume that the quasi-polynomial in (127) has no roots on the imaginary axis. Then is stable, i.e. all of its roots have negative real part if and only if
i.e. the argument of increases as increases from to .
hold, then the characteristic function, and hence the trivial equilibrium point of system (123) is asymptotically stable.
Hence using the characteristic function (126), where
we have for that
It could be seen that
Therefore, we have to show that for each . If it holds, then
and hence by using the Mikhailov criterion we have stability.
Substituting into (159) and multiplying the result by , we get
By using straightforward estimations we obtain that
and hence fulfills for , if the first condition of (155) is satisfied, i.e. if
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.
The characteristic polynomial of (166) is
Since , i.e. the second condition in Theorem 1.10 fulfills, we know from that theorem that if
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 : firstly with , then with and finally with . The Figures 3 and 4 represent the solutions of system (166) with different values of the parameter .
In Figure 3 (above) the parameter is 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 , 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 . 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  Stépán considered the system
where , , , , , and . The characteristic function of system (169) is the quasi-polynomial
In  Freedman and Rao worked with the system
where , , , , , are constants and the characteristic function of (171) is the quasi-polynomial
Let us write these conditions for using the notations of Stépán. In  we can find the condition
in  the condition
Furthermore let us denote the right hand side of the condition in
Since , , , , the numerators of and are equal and
is true following from the positivity of these constants. This means that .
Repeatedly, following from the positivity of the constants , , and we get that the numerators of and are equal, but
hence . 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 a little bit in the following way. In  Stépán used the estimation
but actually this estimation is not sharp for positive , cf. Figure 5.
If we find a tangent line of the function at a certain point , such that this line passes through the origin, i.e. the equation of this line is with a certain , then we can get a better estimation than (179), namely
We can easily determine the constant in the following way: the equation of the searched tangent line at is
where is the function. We would like to find an such that
fulfills, which is true if and only if is a solution of the equation . Let be the solution of this equation, then we find a better linear lower estimation for the function on the positive half-line
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 and to obtain the estimation (164). But if we use higher-order polynomials, we can get a better result, i.e. a better estimation for , such that if satisfies the conditions, then the quasi-polynomial (126) is stable.
and for the delay parameter
then the quasi-polynomial (126) is Hurwitz-stable.
To show that for each we apply the estimations
Then we have
If the conditions (184) are fulfilled, then the coefficient of is positive in the polynomial , moreover if satisfies the condition (185) too, then the discriminant of is negative. Hence with conditions (184) and (185) the inequality is valid for all . 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 .
The characteristic function of (192) is
Let us see what condition gives  for the parameter . The condition
fulfills, hence if
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 , which is greater, than (195). In Figure 6 the phase portrait of the system (192) could be seen with some values of the parameter .
3.1 Stability investigation, independently of the delay
We are going to consider the general form of the characteristic function (126):
where . We assume that if there is not any delay in the system, i.e. , then the trivial equilibrium point is asymptotically stable, which is equivalent to the assumption that the polynomial is Hurwitz stable. We know from  that with this assumption the system (and also its trivial equilibrium point) is delay-independently asymptotically stable if and only if for every the quasi-polynomial has no non-zero real root on the imaginary axis.
In this chapter we will add a condition to the polynomial , and such that the mentioned property on the root of fulfills. In the computations we will follow the idea of .
Firstly, let us multiply the equality by , then we can see that the equivalence
is valid. Let us introduce the notations
With these notations the characteristic function (196) can be written at in the form
then with straightforward calculations we can make the following transformations:
Furthermore we can write the imaginary part in the same way, too:
Since the coefficients , , and in the above matrix are expressed as the linear combination of and , we can write expressions and as follows
Then, by dividing the equalities and by , and introducing a new variable , we obtain
Thus, since , the equation has no real non-zero root for any given if and only of the polynomials and have no common real non-zero root, where , resp. denote the expressions for , resp. . This is equivalent to that or if , then and , where the resultant of the polynomials and is defined as
and the discriminant of a polynomial is Hence we have proved the following statement.
Straightforward calculations show that for all
Thus, the discriminant of is not negative, therefore the stability of system (209) changes at some value of the delay (as we have seen in the previous example).
Figure 7 also shows the changing of the stability of system (209), with the origin is asymptotically stable, but with the origin changes to unstable. The solutions of system (209) with different values of the parameter can be seen on Figure 7. The stability of system (209) changes if the value of the parameter increases.
3.2 Hopf bifurcation
In this subsection we are going to see for which value of the delay could change the stability of the system (123). For this purpose we are going to give conditions on the coefficients , and to obtain the value of the delay at which stability switch may occur. Let us assume that for the conditions of Theorem 1.12 do not fulfill, i.e. for the polynomials and (defined in (206)) the resultant is equal to and the discriminants of or is nonnegative. Furthermore let us assume that is a solution of . Let us denote by the root of the quasi-polynomial (196) that assumes and the characteristic function as a function of the parameter by
Thus, we can determine the derivative of at by the Implicit Function Theorem (cf. ):
where and 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 , and are almost arbitrary polynomials, the fraction could be too complicated, that is why we introduce the following notation:
Since , it is enough to consider the sign of . We are going to use the notations introduced in (198) and along the lines of these we introduce the following notations, too:
(For sake of simplicity replacing by we write for .)
Computing the exact value of , , and we have:
Using the elementary identities
furthermore the Euler formula we can simplify the enumerator of as follows where
Therefore, Hopf bifurcation occurs if holds.
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.
Hale J. Theory of functional differential equations. In: Applied Mathematical Sciences. 2nd ed. Vol. 3. New York-Heidelberg: Springer-Verlag; 1977
Farkas M. Dynamical Models in Biology. New York-Heidelberg: Academic Press; 2001
Gantmacher FR. The theory of matrices. Vol. 1. Translated from the Russian by K. A. Hirsch. Reprint of the 1959 translation
Householder AS. The Theory of Matrices in Numerical Analysis. Reprint of 1964 Edition. New York: Dover Publications, Inc.; 1964
Bissell CC. Stodola, Hurwitz and the genesis of the stability criterion. International Journal of Control. 1989; 50(6):2313-2332
Tóth J, Szili L, Zachár A. Stability of polynomials. Mathematica in Education and Research. 1998; 7(2):5-12
Pontryagin LS. Ordinary differential equations. Translated from the Russian by Leonas Kacinskas and Walter B. Reading, Mass.-Palo Alto, Calif.-London: Counts ADIWES International Series in Mathematics Addison-Wesley Publishing Co., Inc.; 1962
Naulin R, Pabst C. The roots of a polynomial depend continuously on its coefficients. Revista Colombiana de Matemáticas. 1994; 28(1):35-37
Uherka DJ, Sergott AM. On the continuous dependence of the roots of a polynomial on its coefficients. American Mathematical Monthly. 1977; 84(5):368-370
Suter R. Ein kurzer Beweis des Stabilitätskriteriums für kubische Polynome (German) [A short proof of the stability criterion for cubic polynomials]. Zeitschrift für Angewandte Mathematik und Physik. 1983; 34(6):956-957
Orlando L. Su1 problema di Hurwitz relativo aile parti reali delle mdici di un’equationealgebrica. Mathematische Annalen. 1911; 71:233-245
Gantmacher FR. The theory of matrices. Vol. 2. Translated from the Russian by K. A. Hirsch. Reprint of the 1959 translation
Jia L. Another elementary proof of the stability criterion of Liénard and Chipart. Chinese Quarterly Journal of Mathematics. 1999; 14(3):76-79
Kolmanovskiǐ VB, Nosov VR. Stability of functional-differential equations. In: Mathematics in Science and Engineering. Vol. 180. London: Academic Press, Inc.; 1986
Farkas M. Periodic Motions. New York, Inc.: Springer-Verlag; 1994
Younseok C. An elementary proof of the Jury test for real polynomials. Automatica. 2011; 47(1):249-252
Murray JD. Mathematical biology. I. An introduction. New York: Springer-Verlag; 2002
Kovács S, György S, Gyúró N. On an invasive species model with harvesting. In: Mondaini R, editor. Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment. Switzerland AG: Springer, Springer Nature; 2020. pp. 299-334
Asada T, Semmler W. Growth and finance: An intertemporal model. Journal of Macroeconomics. 1995; 17(4):623-649
Liao H, Zhou T, Tang Y. The chaotic region of Lorenz-type system in the parametric space. Chaos, Solitons and Fractals. 2004; 21(1):185-192
Lorenz EN. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences. 1963; 20:130-141
Chen G, Ueta T. Yet another chaotic attractor. International Journal of Bifurcationand Chaos in Applied Sciences and Engineering. 1999; 9(7):1465-1466
Lü J, Chen G. A new chaotic attractor coined. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 2002; 12(3):659-661
Yan Z. Hopf bifurcation in the Lorenz-type chaotic system. Chaos, Solitons and Fractals. 2007; 31(5):1135-1142
Bielczyk N, Bodnar M, Foryś U. Delay can be stabilize: Love affairs dynamics. Applied Mathematics and Computation. 2012; 219:3923-3937
Hals JK, Verduyn Lunel SM. Introduction to Functional-Differential Equations. Berlin, Heidelberg and New York: Springer Verlag; 1993
Churchill RV, Brown JW. Complex Variables and Applications. 8th ed. Boston: McGraw-Hill; 2009
Henrici P. Applied and computational complex analysis. In: Special Functions—Integral Trpansforms—Asymptotics—Continued Fractions. Reprint of the 1977 Original. Vol. 2. New York: Wiley Classics Library. A Wiley-Interscience Publication. John Wiley Sons, Inc.; 1974
Kearns KD. Stability of nuclear reactor systems having time delays. A dissertation submitted’to the faculty of the Department of Nuclear Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate. Tucson, Arizona: College of the University of Arizona; 1971
Freedman HI, Rao SH, V. Stability criteria for a system involving two time delays. SIAM Journal on Applied Mathematics. 1986; 46:552-560
Stépán G. Retarded dynamical systems: Stability and characteristic functions. Pitman Research Notes in Mathematics Series. 1989; 210:159
Hale JK, Infante EF, Tsen FSP. Stability in linear delay equations. Journal of Mathematical Analysis and Applications. 1985; 105(2):533-555
Wu S, Ren G. Delay-independent stability criteria for a class of retarded dynamical systems with two delays. Journal of Sound and Vibration. 2004; 270:625-638
Simon P. Válogatott fejezetek a matematikából (Hungarian) [Topics in Mathematics]. Budapest: Eötvös Kiadó; 2019