Stability Analysis of the Lorenz System Using Hurwitz Polynomials

International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 14, Number 6 (2021), pp. 502-509 © International Research Publication House. http://www.irphouse.com Stability Analysis of the Lorenz System using Hurwitz Polynomials

Fabián Toledo Sánchez1, Pedro Pablo Cárdenas Alzate2 and José Rodrigo González Granada3 1,2,3 Department of Mathematics and GEDNOL, Universidad Tecnológica de Pereira, Pereira, Colombia.

Abstract Among these fundamental studies is the stability analysis; this stability analysis can be performed using the results of In the qualitative study of differential equations, we can Lypaunov, who presented two methods to determine stability analyze whether or not small variations or perturbations in the that solve particular problems according to the structure of the initial conditions produce small changes in the future, this differential equations or using Hurwitz type polynomials. intuitive idea was formalized by Lyapunov in his work "The General Problem of Motion Stability". The stability analysis of If the nonlinear ordinary differential equation is expressed in a system of differential equations can be performed using the the form 푥̇ = Ax, then the problem of analyzing the stability methods proposed by Lyapunov or using criteria to obtain becomes a problem of algebraic type, since it is only enough to Hurwitz type polynomials, which provide conditions to analyze know the roots of the characteristic polynomial associated to the dynamics of the system by studying the location of the roots the matrix 퐴 which correspond to the eigenvalues and to of the characteristic polynomial associated to the system. In this observe if these have negative real part; if the above occurs it paper we present a stability analysis of the Lorenz system using is said that the system 푥̇ = Ax is asymptotically stable. If this stability criteria to obtain Hurwitz type polynomials. polynomial has the above mentioned characteristic it is said to be a Hurwitz polynomial [6], [7]. Keywords: Stability of differential equations, Hurwitz polynomials, Hurwitz criterion, Lorenz system. Therefore, a relevant fact to analyze the stability of the system 푥̇ = Ax corresponds to establish criteria to determine when the

characteristic polynomial associated to the matrix 퐴 is Hurwitz, I. INTRODUCTION i.e., if all its roots have negative real part. In [8] a series of criteria are presented to obtain Hurwitz type polynomials, these The stability analysis of systems of ordinary differential criteria present some equivalences in their formulation, among equations is the object of study in various mathematical models them the Routh-Hurwitz criterion, the Lienard-Chipart applied to a varied branch of the exact sciences and control conditions, the Hermite-Biehler theorem, the stability test and theory, where the aim is to provide necessary and sufficient the Routh algorithm [9]. conditions so that in a given problem with established initial conditions, solutions close to this initial value remain close Thus, the objective of this paper is to analyze the stability of throughout the future or in a given case tend to the equilibrium the Lorenz system using the Routh-Hurwitz stability criteria solution. This notion of closeness and proximity for small and the Routh algorithm. For this purpose, four sections are variations in the initial conditions was initially worked on by presented: in the first one, the Routh-Hurwitz stability criteria outstanding mathematicians such as Lagrange and Dirichlet, and the Routh algorithm are deduced; in the second section, a but it was not until 1892 when the Russian mathematician and stability analysis of the Lorenz system is performed using the physicist Aleksandr Lyapunov laid the foundations for this criteria presented to obtain Hurwitz type polynomials and concept of proximity in his doctoral thesis entitled "The finally, the results are analyzed by means of simulations General Problem of Motion Stability" [1]. performed in MATLAB using the ode45 function for the solution of the system of nonlinear differential equations. The The study of the dynamics of these sensitive variations in the simulations validate the results obtained on the stability of the initial conditions is the fundamental work in the qualitative Lorenz system. analysis of ordinary differential equations. The importance of such sensitive variation is observed in the Lorenz system, this nonlinear model that described the movement of air masses in the atmosphere gave way to a branch of mathematics called II. HURWITZ POLYNOMIALS chaos theory, because Lorenz, studying the weather patterns, Let began to observe unusual behaviors to those that had been established that should happen. Lorenz using a numerical 풙̇ = 퐴풙 (1) simulation used 0.506 as an initial data as a less accurate be a system of linear or nonlinear ordinary differential approximation of the data 0.506127, obtaining as a result, equations, where 퐴 is a square matrix and 풙 is a vector. The according to two tabulations, two completely different climate stability of the system (1) at its equilibrium point can be scenarios [2], [3]. This small variation or perturbation in their determined by performing a study of the eigenvalues of the initial values yielded completely different climatological associated matrix 퐴. This algebraic study establishes that if the results. From the moment Lorenz formulated the problem with roots of the characteristic polynomial associated to the matrix this question, various characteristics of the system began to be A have negative real part, then we conclude that the system (1) studied, such as regions of stability, attractor basins, is asymptotically stable. bifurcations, chaos, geometric aspects, among others [4], [5].

502 International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 14, Number 6 (2021), pp. 502-509 © International Research Publication House. http://www.irphouse.com

Now, the problem of determining the stability in a system of This matrix is constructed as follows: differential equations from the study of the roots of the  In the first row are the coefficients of the polynomial characteristic polynomial of 퐴 associated to the linear or (3) with odd location starting with 푎1. nonlinear system, translates into the task of finding necessary  In the second row are the coefficients of the and sufficient conditions for which all the roots of this polynomial (3) with even location starting with 푎0. polynomial are located in the left half of the complex plane, i.e.,  The elements of each subsequent row are formed so if they have negative real part. that the component ℎ푖푗 is given by: This way of obtaining polynomials of this type with negative real part was initially proposed by the physicist James Maxwell 푎 푠푖 0 < 2푗 − 푖 ≤ 0 in 1868, who presented a solution to this problem for ℎ = { 2푗−푖 푖푗 polynomials of degree 3, then in 1877, the Canadian 0 푖푛 표푡ℎ푒푟 푐푎푠푒 mathematician Edward Routh presented an algorithm that As a result of the construction, the coefficients 푎1, 푎2, 푎3, solved the problem in a more general way providing explicit … , 푎푛 are on the main diagonal of the matrix, and all the conditions for polynomials up to degree 5 and later it was the elements of the last column are null, except the last element German mathematician Adolf Hurwitz in 1895 who presented which is 푎푛. The matrix ℋ is called Hurwitz Matrix. a properly analytical solution to the problem [6]. Routh-Hurwitz theorem. The polynomial (3), with its Definition 1: A polynomial with real coefficients is said to be positive leading coefficient (푎0 > 0), is a Hurwitz polynomial Hurwitz if all its roots have a negative real part, that is, if all its if and only if all the diagonal principal minors of the Hurwitz roots lie in ℂ−, the left half-plane of the complex plane, matrix ℋ are positive [10]. ℂ− = { 푎 + 푏푖 ∶ 푎 < 0 } (2) The principal diagonal minors of the matrix (4) are given by the following determinants,

푎1 푎3 푎5 II.I Criteria for obtaining Hurwitz polynomials 푎1 푎3 Δ1 = ∣ 푎1 ∣, Δ2 = | |, Δ3 = |푎0 푎2 푎4|, 푎0 푎2 Let 0 푎1 푎3

푛 푛−1 2 푎1 푎3 푎5 푎7 푃(휆) = 푎0휆 + 푎1휆 + ⋯ + 푎푛−2휆 + 푎푛−1휆 + 푎푛 (3) 푎0 푎2 푎4 푎6 Δ4 = | |, ⋯ , Δ푛 = 푎푛 ∙ Δ푛−1. be the characteristic polynomial associated to the matrix 퐴 of 0 푎1 푎3 푎5 the system (1). 0 푎0 푎2 푎4 In the search for algorithms to determine the location of the roots of the polynomial (3) the following question was asked: II.I.II Routh criterion how to determine if the polynomial (3) is Hurwitz? For the coefficients of the polynomial (3) the Routh This question was studied by several mathematicians and arrangement is constructed as follows, physicists, among the first who proposed to solve this type of problem was the Austrian engineer A. Stodola who at the end 푎0 푎2 푎4 푎6 ⋯ of the 19th century was interested in the problem of finding 푎1 푎3 푎5 푎7 ⋯ conditions under which all the roots of a polynomial had a negative real part; but in 1895 it was Hurwitz who presented a 푏0 푏1 푏2 푏3 ⋯ (5) 푐 푐 ⋯ solution to the previous question based on the work of Hermite. 푐0 푐1 2 3 푑 푑 ⋯ 푑0 푑1 2 3 There are several criteria to obtain Hurwitz type polynomials. ⋮ ⋮ ⋮ ⋮ ⋱ In this paper we only briefly present the Routh-Hurwitz criterion and the Routh algorithm.  In the first row are the coefficients of the polynomial (3) with even location starting with 푎0. II.I.I Routh-Hurwitz criterion  In the second row are the coefficients of the polynomial To present the Routh-Hurwitz criterion we first construct the (3) with odd location starting with 푎1. following matrix from the coefficients 푎0, 푎1, … , 푎푛 of the  The elements of each subsequent row are formed polynomial (3): according to the following algorithm

푎 푎 푎1 푎3 푎5 푎7 ⋯ 0 0 2 푏0 = 푎2 − 푎3 , 푏1 = 푎4 − 푎5 , ⋯ 푎 푎 푎 푎 ⋯ 0 푎 푎 0 2 4 6 1 3 푎 푎 푎 ℋ = 0 1 2 3 ⋯ 0 (4) 푎1 푎3 푎2 푎 푎 ⋯ 0 0 4 6 푐0 = 푎3 − 푏1 , 푐1 = 푎5 − 푏2 , ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ 0 푏0 푏1 [ 0 0 0 0 ⋯ 푎푛] 푏0 푏1 푑0 = 푏1 − 푐1 , 푑1 = 푏2 − 푐2 , ⋯ 푐0 푐1

⋮ ⋮

503 International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 14, Number 6 (2021), pp. 502-509 © International Research Publication House. http://www.irphouse.com

Routh's theorem: From (7) we have that the equilibrium points are given at the origin 풪 = (0,0,0) for all values of 휎, 훾 and 훽, and at the The number of roots of the polynomial 푃(휆) in the right half- plane of the complex plane is equal to the number of sign points 풟1,2 = (± √훽(훾 − 1), ± √훽(훾 − 1) , 훾 − 1) for 훾 > 1. variations of the first column in Routh's array (5). Routh criterion: III.I.I Stability of the origin The polynomial 푃(휆) is Hurwitz polynomial if and only if Stability in 풪 = (0,0,0) is obtained by linearizing the flux in 풪, when performing Routh's array (5) all values in the first i.e: column are nonzero of the same sign [11]. −휎 휎 0 풥(0,0,0) = [ 훾 −1 0 ] 0 −훽 III. LORENZ SYSTEM 0 Where 풥 is the Jacobian associated to the system (6). The The American mathematician and meteorologist Edward characteristic polynomial is given by: Lorenz according to his studies on atmospheric phenomena built a mathematical model that would give rise to an important 풫(휆) = 휆3 + (훽 + 휎 + 1)휆2 + [훽(휎 + 1) + 휎(1 − 훾)]휆 (8) branch of dynamical systems called chaos theory whose + 훽 휎 (1 − 훾) foundations appear in his article Deterministic Non-periodic Flow [12] published in 1963 describing the so-called butterfly To analyze the nature of the eigenvalues that will determine the effect. This nonlinear model is given by: stability in 풪, we use the Hurwitz criterion. 푧̇ = 휎 (푦 − 푥) Thus, the Hurwitz matrix associated with the polynomial 풫(휆) (8) for 푎 = 1, 푎 = 훽 + 휎 + 1, 푎 = 훽(휎 + 1) + 휎(1 − 훾) {푦̇ = 푥 (훾 − 푧) − 푦 (6) 0 1 2 and 푎 = 훽 휎 (1 − 훾), is given by: 푧̇ = −훽푧 + 푥푦 3 훽 + 휎 + 1 훽 휎 (1 − 훾) where 휎, 훾 and 훽 are positive parameters and 휎 ∶= Prandtl ℋ = [ ] 1 훽(휎 + 1) + 휎(1 − 훾) number, 훾 ∶= Rayleigh number and 훽 is a proportionality constant. In Lorenz equations usually the values of 휎 = 10, Applying Hurwitz's theorem, we have that the polynomial 8 훽 = are taken and 훾 can take any positive value. When 훾 = 풫(휆) (8) is Hurwitz, if it is satisfied that: 3 28 the system (6) presents a chaotic behavior whose Δ1 = ∣ 훽 + 휎 + 1 ∣ > 0 simulation is presented in Figure 1, where a numerical 훽 + 휎 + 1 훽 휎 (1 − 훾) approximation of a solution with two initial conditions close to Δ2 = | | > 0 the origin is given, with this we observe that a small 1 훽(휎 + 1) + 휎(1 − 훾) perturbation in the initial conditions produces large changes in Δ1 is positive because the parameters 훽 and 휎 are positive, we their respective trajectories [13]. only need Δ2 to be positive, that is: (훽 + 휎 + 1)[ 훽(휎 + 1) + 휎(1 − 훾)] − 훽 휎 (1 − 훾) > 0 (9) This is true for (휎 + 1)[훽2 + 훽(휎 + 1) + 휎(1 − 훾)] > 0 and for all 훾 < 1. Therefore, the polynomial 풫(휆) (8) is Hurwitz if and only if 훾 < 1, and indeed the system is asymptotically stable in 풪 if 훾 < 1.

III.I.II Equilibrium point stability 퓓ퟏ,ퟐ Figure. 1. Lorenz attractor. For 훾 > 1 we have the equilibrium points,

풟1 = ( √훽(훾 − 1), √훽(훾 − 1) , 훾 − 1) and III.I Stability analysis of the Lorenz system 풟 = (− √훽(훾 − 1), − √훽(훾 − 1) , 훾 − 1) To perform a stability analysis of the Lorenz system (6) we 2 start by establishing its equilibrium points, i.e., the 푥̃ such that Linearizing the flow in 풟1 or 풟2 we have that, 푓(푥̃) = 0, that is: −휎 휎 0 휎 (푦 − 푥) = 0 1 −1 풥(풟1,2) = [ − √훽(훾 − 1) ] 푥 (훾 − 푧) − 푦 = 0 (7) √훽(훾 − 1) √훽(훾 − 1) −훽 −훽푧 + 푥푦 = 0 whose characteristic polynomial 풫(휆) is given by:

504 International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 14, Number 6 (2021), pp. 502-509 © International Research Publication House. http://www.irphouse.com

풫(휆) = 휆3 + (훽 + 휎 + 1)휆2 + 훽(휎 + 훾)휆 + 2훽 휎 (훾 − 1) (10) As in the previous case, to analyze the nature of the eigenvalues of the polynomial (10) that will determine the stability in 풟1 and 풟2 we make use of Routh's algorithm for para 푎0 = 1, 푎1 = 훽 + 휎 + 1, 푎2 = 훽(휎 + 훾) and 푎3 = 2훽 휎 (훾 − 1), thus, 1 푏 = 훽(휎 + 훾) − ∙ 2훽 휎 (훾 − 1) 0 훽 + 휎 + 1 and Routh's arrangement (5) is given by 1 훽(휎 + 훾) 훽 + 휎 + 1 2훽 휎 (훾 − 1) (11) 2훽 휎 (훾 − 1) 훽(휎 + 훾) − 훽 + 휎 + 1 Figure. 2. 훾 = 0.5 Then, by Routh's theorem we must analyze the signs of the first  If 훾 = 1, the equilibrium point 풪 presents a bifurcation column of (11). Since 푎0 and 푎1 are positive then it only called the hairpin. remains to analyze the sign of 푏 as a function of the variable 0  If 훾 > 1 , the origin is unstable. Figure 3 shows that parameter 훾 . Let us note that if 푏0 is positive then the polynomial (10) is Hurwitz and indeed we have stability at the stability is lost at the origin 풪. equilibrium points 풟1 and 풟2, therefore, let us see for which value of 훾, 푏0 is positive, that is: 2훽 휎 (훾 − 1) 푏 = 훽(휎 + 훾) − > 0 0 훽 + 휎 + 1 This is true if 휎(휎 + 훽 + 3) 훾 < 휎 − 훽 − 1 휎(휎+훽+3) Now, if 훾̃ = [7], then we conclude that 푏 is positive 휎−훽−1 0 when it is satisfied that 훾 < 훾̃ and indeed by Routh's theorem as in the first column of (11) the values of 푎0, 푎1 and 푏0 are positive for the above constraint it follows that the polynomial (10) is Hurwitz and therefore we have that 풟1 and 풟2 are asymptotically stable when 1 < 훾 < 훾̃ . Figure. 3. 훾 = 1.8

Furthermore, according to the characteristic of the eigenvalues IV. RESULT AND DISCUSSION of the polynomial (8), i.e. In this section we will analyze the stability results obtained −(휎 + 1) ± √(휎 + 1)2 − 4휎 (1 − 훾) 휆 = −훽, 휆 = previously together with their respective simulation performed 1 2,3 2 in MATLAB using the ode45 function for the solution of the system of nonlinear differential equations. where 휆1 and 휆3 are negative and 휆2 is positive, the origin presents an unstable variety 푊푢(0) one-dimensional and a stable variety 푊푠(0) two-dimensional (see Figure 4 (a-c)). If 훾 IV.I Stability at 퓞 grows to about 13.926 the spirals touch the stable variety of the origin and thus this point becomes homoclinic, i.e., there is an  If 0 < 훾 < 1 , the origin 풪 is asymptotically stable, orbit that starts and ends at the origin, as seen in figure 4(d). moreover, the origin is globally stable, i.e., all trajectories tend to the origin, as seen in Figure 2.

505 International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 14, Number 6 (2021), pp. 502-509 © International Research Publication House. http://www.irphouse.com

(a) 0 < 훾 < 1 (b) 훾 = 1

IV.I Stability at 퓓ퟏ,ퟐ (as shown in Figure 5. (d-i)) i.e., there comes a time when the trajectories vary their stability course between 풟1 and 풟2. If 1 < 훾 < 훾̃ , the equilibrium points 풟1 and 풟2 are 8 asymptotically stable, if 휎 = 10 and 훽 = then 훾̃ ≈ 24.74 In Figure 5. two trajectories with initial conditions near the 3 origin 풪 were taken. In Figure 5. (a-c) the trajectories tend to and indeed 1 < 훾 < 24.74 . It is important to note that the equilibrium point 풟 and just at about 13.926 the trajectories according to numerical analysis for initial conditions near the 1 change their course towards the equilibrium point 풟 , as origin, if 1 < 훾 < 13.926 then 풟 is stable (as seen in Figure 2 1 observed in the transition from Figure 5. (c) to Figure 5. (d). 5. (a-c)) and if 13.926 < 훾 < 훾̃ ≈ 24.74 then 풟2 is stable

(a) 훾 = 7 (b) 훾 = 10

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(e) 훾 = 15 (f) 훾 = 18

(g) 훾 = 22 (h) 훾 = 24

(i) 훾 = 24.5 (j) 훾 = 24.74

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(k) 훾 = 25 (l) 훾 = 27

(m) 훾 = 30 (n) 훾 = 35 Figure. 5. Trajectories for 1 < 훾 < 훾̃, 훾 = 훾̃ and 훾 > 훾̃.

For values of 훾 greater than 13.926 the trajectories tend to the performed in MATLAB, for the values of 훾 yielded by the equilibrium point 풟2 as seen in Figure 5. (d-i). Routh-Hurwitz and Routh criterion. If 훾 = 훾̃, one has a Hopf-type bifurcation where closed periodic orbits are formed near the equilibrium points 풟 and 풟 . 1 2 ACKNOWLEDGEMENTS As 훾 approaches the value 훾̃ = 24.74 , the trajectories lose stability, i.e., the behavior of the trajectories as time elapses This work was supported by the GEDNOL Group Research at cannot be predicted, as seen in the transition in Figure 5. (i-k). the Universidad Tecnológica de Pereira - Colombia and would If 훾 > 훾̃, the equilibrium points 풟 and 풟 are unstable. like to thank the referee for his valuable suggestions that 1 2 improved the presentation of this paper. For values of 훾 higher than 24.74 the trajectories are unstable, i.e., in spite of a small variation in the initial conditions, these REFERENCES trajectories have different behaviors in the future, as shown in Figure 5(j-n). [1] A. . Lyapunov, “The General Problem of the Stability of motion,” vol. 6, no. 11, pp. 951–952, 1992.

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