FOR ONE SYSTEM OF DIFFERENTIAL EQUATIONS TAKEN AS DUAL OF THE LORENZ SYSTEM

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1 Математички Билтен ISSN X (print) Vol 4(LXVI) No ISSN (online) 4 (37-44) UDC: 5793:5797 Скопје, Македонија FOR ONE SYSTEM OF DIFFERENTIAL EQUATIONS TAKEN AS DUAL OF THE LORENZ SYSTEM Boro M Piperevski Abstract This paper looks at and goes through Lorenz sstem and his dual sstem of differential equations Within analtics we have come to some commonalities in both sstems INTRODUCTION Here we have the Lorenz sstem: x x x r z, z x bz,, rb, In literature [, 7, 8] it is known that sstem () has three singularities (fixed point, equilibrium point) in the points O(,,), O( b r, b r, r ), O( b r, b r, r ) for r, and one singularit in point O (,,), for r In accordance with the theor of stabilit [,, 3, 5, 7, 8], due to importance sstem () stabilit examination for singularities we calculate appropriate characteristic equations The characteristic equation, for sstem (), is given with ( b)[ ( r)], (), () Mathematics Subject Classification Primar: 37-G35, 34-C3, Secondar 34-K8 Ke words and phrases sstem of differential equations, dual sstem of Lorenz sstem, singularities (fixed point, equilibrium point), characteristic equations 37

2 38 Boro M Piperevski for r in singularit O (,,) The characteristic equation is given with for r in singularities b b r br 3, (3), O ( b r, b r, r ) O ( b r, b r, r ) Lemma Let an algebraic equation of third degree is given 3 x a x ax a3, a, a, a3 R This equation has two conjugated pure imaginar roots if following condition is met [] аа а3 (4) MAIN RESULT Let us bring transformation shift defined with x x ix, i, z z iz (5) where x, x,,, z, z are real functions from real argument t, i With direct substitute (5) in () we get sstem of differential equations x z x z x rx x z x z x x bz x rx x z x z x x bz Lemma The sstem of differential equations (6) has three singularities in O (,,,,, ), O (,, r, b r, b r, ), for r, and three singularities O (,, r, b r, b r, ), (6)

3 FOR ONE SYSTEM OF DIFFERENTIAL EQUATIONS 39 * O(,,,,, ), O ( b r, b r ), r,,, ), O * ( b r, b r, r,,,), for r Proof B solving the algebraic equations sstem x rx x z x z x x bz x rx x z x z x x bz, we get the coordinates of singularities Lemma 3 For a sstem of differential equations (6) in common singularit O (,,,,, ) appropriate characteristic equation is given with b for r, and in remaining singularities ( r), (7) O(,, r, b r, b r, ), O(,, r, b r, b r, ), * * O( b r, b r ), r,,, ), O( b r, b r, r,,,) appropriate characteristic equation is given with b b r br 3, (8) in both cases r and r Proof Within calculating the matrix of Jacobi Ј О r b r b and the corresponding determinant

4 4 Boro M Piperevski r b, r b in common singularit O (,,,,, ), for r, we get characteristic equation (7) With calculating the matrix of Jacobi * ЈO ( ) b r b r b r b b r b r b r b and the corresponding determinant b r b r b r b * in singularit O ( b r, b r ), r,,, ) ristic equation (8) With calculating the matrix of Jacobi Ј b r b r b r b, for r, we get character-

5 FOR ONE SYSTEM OF DIFFERENTIAL EQUATIONS 4 br b r b b r br b r b r b and the corresponding determinant br b r b b r br b r b r b in singularit O (,, r, b r, b r, ), for r, we get characteristic equation (8) We get the same equation (8) in other singularities for r, and for r O * ( b r, b r, r,,,), O (,, r, b r, b r,), Note Let us note that the characteristic equations (7) and (8) are in fact squares of the characteristic equations () and (3) for the Lorenz sstem

6 4 Boro M Piperevski 3 DUAL SYSTEM OF THE LORENZ SYSTEM Let us now look at the sstem x x rx x z z x bz gotten from the sstem (6) as his subsstem for x,, z (9) Definition The sstem (9) is called a dual sstem of the Lorenz sstem Theorem The sstem (9) has three singularities in the points O(,,), O( b r, b r, r ), O( b r, b r, r ), when limitation is r, and one singularit in point O (,,), when limitation is r Proof From the algebraic equations sstem x, we get singularities coordinates rx x z, x bz, Theorem For sstem (9), appropriate characteristic equation in singularit O (,,) is given with ( b)[ ( r)], () for r For r all roots of this equation () are negative The characteristic equation in other singularities is given with O( b r, b r, r ), O( b r, b r, r ) b b r br 3, () for r The characteristic equation () does not have purel imaginar roots Proof The matrix of Jacobi of sstem (9) in singularit is

7 FOR ONE SYSTEM OF DIFFERENTIAL EQUATIONS 43 J ( O) r b This means that the characteristic equation () is achieved with calculation of the determinant Ј ( O) Е r b Equation of the roots () are For r we get 4 r b,,3 4 r, where from we have that all three roots are negative The Jacobi matrix of the sstem (9) in the singularit is O ( b r, b r, r ), J O b( r) b( r) b( r) b This means that the characteristic equation () is achieved with calculation of the determinant Ј O Е b( r) b( r) b( r) b We get the same equation () as well in the other singularit

8 44 Boro M Piperevski O ( b r, b r, r ) According to the Lemma and the condition (4), with application of the characteristic equation () we get the condition bb r br Due to the limitation r, the condition is not satisfied because the left side is alwas a positive number, and the right side is alwas a negative number Note Let us note that characteristic equations () and () are the same with appropriate characteristic equations () and (3) for Lorenz sstem () Along with this it is important to point that for this characteristic equation the condition r is valid, while for them at Lorenz sstem the condition r is valid References [] J E Marsden, M McCracken: The Hopf Bifurcation and its Applications, Springer Verlag, New York 976 [] G Ioss, D D Joseph: Stabilit and Bifurcation Theor, Springer Verlag, New York, Heidelberg, Berlin98 [3] Arnold V I: Ordinar Differential Equations, MIT978 [4] ELInce: Ordinar Differential Equations, Dover Publications INC New York 956 [5] Л С Понтрягин: Обыкновенные дифференциальные уравнения, Наука, Москва97 [6] И Г Петровский: Лекции по теории обыкновенных дифференциальных уравнений, Наука, Москва, 97 [7] Schroeder M: Fractals, Chaos, Power Laws, WHFreeman and Co, New York99 [8] R Barrio, F Blesa, SSerrano: Behavior patterns in multiparametric dnamical sstems: Lorenz model, International Journal of Bifurcation and Chaos, Vol, No 6 () 39 Facult of Electrical Engineering and Information Technologies, Universit Ss Cril and Methodius, Skopje, Macedonia, address: borom@feitukimedumk