Two Limit Cycles in a Two-Species Reaction

Two Limit Ccles in a Two-Species Reaction Brigita Ferčec 1 Ilona Nag Valer Romanovski 3 Gábor Szederkéni 4 and János Tóth 5 The Facult of Energ Technolog Krško 1 Center for Applied Mathematics and Theoretical Phsics Universit of Maribor Facult of Electrical Engineering and Computer Science Universit of Maribor Facult of Natural Science and Mathematics Universit of Maribor 3 Facult of Information Technolog and Bionics Pázmán Péter Catholic Universit 4 Department of Analsis Budapest Universit of Technolog and Economics 5 Miklós Farkas Seminar on Applied Analsis September 7 018 Contents 1 Introduction 1.1 Eistence of periodic trajectories 1. Eclusion of periodic trajectories 1.3 Limit ccles 1.4 The model with a large and a small limit ccle Methods to be used.1 Poincaré compactification. Homogeneous directional blow-up 3 The model with two limit ccles 3.1 Preparation for the analsis of the model 3. Phase portrait on the Poincaré disk 3.3 The big and the small limit ccle 3.4 Plotting the limit ccles 4 References 1 Introduction 1.1 Eistence of periodic trajectories - Eotic behavior of chemical reactions: oscillation multistabilit multistationarit or chaos - Oscillator chemical reactions ma also form the basis of periodic behavior in biological sstems - The second part of the 16th problem of David Hilbert (1900): find the number of limit ccles of twodimensional autonomous polnomial differential sstems. - The problem seems to be ver difficult even in the case of two-dimensional kinetic differential equations. - Schlomiuk and Vulpe (01): in the class of quadratic differential equations the number of different phase portraits is estimated to be more than 000

018-09-7.nb - Frank-Kamenetsk (1947): modelling the oscillation of cold flames The Lotka-Volterra equation ' k 1 k ' k k 3 was reinterpreted as the induced kinetic differential equation of the reaction quadrant X k 1 X X Y k Y Y k 3 0 Nonlinear first integral conservative oscillations i.e. closed trajectories in the first - The Belousov-Zhabotinsk reaction (Belousov: 1958 Zhabotinsk: 1964) is an oscillating chemical reaction. https:en.wikipedia.orgwikibelousovzhabotinskreaction - Hsü (1976): the Oregonator model of the BZ reaction has periodic solutions b a theorem on Andronov- Hopf bifurcation - Field Kőrös and Noes (197): Oscillations in chemical sstems 1. Eclusion of periodic trajectories Application of the theorem b Bendison or b Bendison and Dulac: - Wegscheider reaction: The induced kinetic differential sstem ' k 1 k 1 k k : f ' k 1 k 1 k k : f of the reversible reaction (X k 1 k 1 k Y X Y) has no periodic trajector. k - Bautin (1954): within the class of equations ' a b c ' d e f onl those can have a periodic solution which are of the Lotka-Volterra form (Bendison-Dulac theorem) - Póta-Hanusse-Tson-Light theorem (proofs: Póta with a Dulac function 1983; Schuman and Tóth 003): Among two-species second-order reactions the onl oscillator reaction is the Lotka-Volterra model.

018-09-7.nb 3 Suppose that the coefficients of the equation ' a b c d e f ' A B C D E F obe the inequalities 1 0 c e f A D F the equation is kinetic 0 a C no steps like X 3 X or Y 3 Y occur 3 at most one of b and B is positive no steps like XY XY or XY X Y occur Then the onl equation to have periodic solutions is of the Lotka-Volterra form specificall limit ccles cannot arise. Interesting question: is condition (1) enough to eclude the emergence of limit ccles? -Escher (1980 1981): chemical eamples with two species and second-order reactions with even more than one limit ccles but long product complees are allowed Both conservative oscillations and limit ccles: Y X 3 X X 0 Y X Y 0 1.3 Limit ccles Prigogine and Lefever (1968): the Brusselator model with a limit ccle (a Hopf bifurcation emerges) 1 b a 0 X Y X Y 3 X 1 ' 1 b 1 a ' b a Gra and Scott (1986): the Autocatalator model 0 k 0 Y k 3 X k 0 X Y k 1 3 X ' k 3 k k 1 ' k 0 k 3 k 1 k 1 Erle (1998): If 0 m' m; 0 Β n'; Α 0 then for the reaction m X n Y m' X n' Y k 1 ' Α X k' k 0 k3' there eist reaction rate coefficients for which the reaction has an asmptoticall orbitall stable closed orbit. k 3 Β Y Erle (000): Nonoscillation in closed reversible chemical sstems Schnakenberg (1979): For ehibiting limit ccle behavior a two-species reaction has to consist of at least three reaction steps among which one must be autocataltic of the tpe X Y 3 X. The possible candidates are those whose stationar state is an unstable focus. Császár Jicsinszk and Turáni (198): The used necessar conditions to construct candidate reactions with limit ccles. The have shown numericall that some of the reactions seem to have limit ccles. Schlosser and Feinberg (1994): a graph theoretical necessar condition of periodicit and multistationarit

4 018-09-7.nb 1.4 The model with a large and a small limit ccle The dnamical sstem investigated in [1] comes from a chemical model published b Császár et al. (198): 0 K 1 V U K 0 V K 3 U U K 4 V U V K 5 3 U where K i 0 i 1... 5 are the reaction rate coefficients. The induced kinetic differential equations are u' K u K 4 u K 3 v K 5 u v v ' K 1 K 4 u K 3 v K 5 u v B. Ács G. Szederkéni Zs. Tuza and Z. A. Tuza (016): precisel 17160 reaction graphs with different structure can produce eactl the same dnamical behavior: the induce the same mass action tpe kinetic differential equations. However the computed structures could not be used to show important dnamical properties of the model (e.g. the eistence of positive equilibria or the boundedness of solutions). Methods to be used.1 Poincaré compactification The Poincaré compactification is one of he tools to stud the behavior of the trajectories of a planar differential sstem near infinit. Each polnomial vector X P Q of degree d of the sstem ' P ' Q can be analticall etended to the Poincaré sphere S z R 3 : z 1. This can be done b central projection of the points M 1 R 3 onto S. Northern hemisphere: H z S : z 0; M '' 1 1 1 1 H Southern hemisphere: H z S : z 0; Equator: S 1 z S : z 0 M ' 1 1 1 1 H The finite points of the plane are projected to the northern hemisphere and the southern hemisphere and the infinite points to the equator.

018-09-7.nb 5 For the investigation of the points on the sphere si local charts will be used: U i s S : s i 0 i 1 3 V i s S : s i 0 i 1 3 where d denotes the degree of the sstem. V i distinguishes from U i b the factor 1 d1. The finite points correspond to the charts U 3 V 3 respectivel. The infinite points correspond to the charts U 1 U V 1 V where v 0. It is enough to consider the points on U 1 v0 and U 00 to understand the behavior of the infinite points. Poincaré disk: the projejction of the points of the northern hemisphere of S to the equator. Eample 1 ' ' Singular point: 0 0 it is a saddle. Degree: d 1. Projection to U 1 : 1 v u v u v 1 The transformed sstem: u' u v ' v Singular point at U 1 : 0 0 it is a stable node at infinit. Since d is odd then the origin of V 1 is also a stable node.

6 018-09-7.nb Projection to U : u v 1 v u v 1 The transformed sstem: u' u v ' v Singular point at U : 0 0 it is an unstable node at infinit. The origin of V 1 is also an unstable node. The phase portrait on R : The phase portrait on the Poincaré disk: 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 Eample ' ' Singular points: 0 0: saddle (-1-1): center (it can be seen with the first integral H 1 3 3 1 3 3 ). Degree: d. Projection to U 1 : 1 v u v u v 1 The transformed sstem: u' 1 u 3 u v v ' u v v Singular point at U 1 : 1 0 (we need onl the case v 0) it is an unstable node at infinit. Since d is even the diametricall opposite point is a stable node in V 1. Projection to U : u v 1 v u v 1 The transformed sstem: u' 1 u 3 u v v ' u v v Singular point at U : the onl possibilit is u 0 v 0 but 0 0 is not a singular point so there are no additional infinite singular points. The phase portrait on R : The phase portrait on the Poincaré disk: 1 0 1 1 0 1

018-09-7.nb 7. Homogeneous directional blow-up Blow-up: transformation of the variables so that the behavior near a degenerate singular point is possible to determine. Two tpes of blow-ups: ' P P m... ' Q Q m... polar blow-up : singular point circle directional blow-up : singular point straight line P m Q m : homogeneous polnomials of degree m N and the dots mean higher order terms We assume that the origin is a singular point since m 0. The characteristic function to determine the direction of the blow-up: F Q m P m If F 0 then P m W m1 and Q m W m1 where W m1 0. The angle φ 0 Π is the singular direction if the factor of W m1 is of the form v where v tanφ. The blow-up in the direction is the transformation z and the transformed sstem is Q z z P z ' P z z' The singularit is transformed into the line 0 called the eceptional divisor. The appearing common factor m1 sometimes m needs to be cancelled. The nd and 3rd quadrants are swapped. 0 0 z 0 0 0 z 0 0 0 z 0 0 0 z 0 The blow-up in the direction is the transformation z and the transformed sstem is P z z Q z z' ' Q z The singularit is transformed into the line 0. The 3rd and 4th quadrants are swapped. Eample 1 ' 1 ' 3 (1) The characteristic function: ' 1 4 P ' 3 Q 0 F Q P 0 F 0 if 0 directional blow up Blow-up in the direction: z

8 018-09-7.nb The transformed sstem: ' 1 z z' z 1 z () Cancelling : ' 1 z z' z 1 z (3) The origin is transformed into the line 0. On this line the onl stationar point is 0 0 with eigenvalues 1 1 0 0 is a saddle. 3 1 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 0.5 0.5 0.5 1.0 1.0 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 Eample ' 3 3 ' 3 (1) The characteristic function: F Q 1 P 1 0 F 0 if 0 blow up in the direction : z The transformed sstem: ' z 3 z 3 z 3 z' z 3 z 3 z 4 () The origin is still degenerate so an additional blow-up needs to be done. The characteristic function: F Q z z P z z z z 4 z F 0 if 0 blow up in the z direction : t z z z The transformed sstem: t ' t z 4 t 6 t z t z 4 z' z t 3 t z t z 4 (3) Cancelling z: t ' t 4 t 6 t z t z 4 z' z t 3 t z t z 4 (4) The origin is transformed into the t ais. Here (on the line z 0) the stationar points are 0 0 0 0. The origin is a saddle with eigenvalues 4.

018-09-7.nb 9 3 The model with two limit ccles 3.1 Preparation for the analsis of the model The number of parameters in the following sstem can be decreased b two: u' K u K 4 u K 3 v K 5 u v v ' K 1 K 4 u K 3 v K 5 u v Let ut a t Τ vt b t Τ where a b Τ are constants u' t aτ t Τ v ' t bτ t Τ. The new sstem is: ' K a K 4 Τ Τ b K 3 a Τ a b K 5 Τ ' K 1 b Τ a K 4 K 3 a K 5 b Τ Τ Τ Let the coefficient of and be equal to 1 a b Τ a K 5 and a K 4 K 5. If k 1 K 1 K 5 K 4 3 k K K 5 k 3 K 3 K 5 K 4 K 4 then the new sstem is:

10 018-09-7.nb ' k k 3 ' k 1 k 3 (1) which can be obtained from the original one if K 4 K 5 1 and the notation is changed appropriatel. 3. Phase portrait on the Poincaré disk Theorem 1. For an positive values of parameters k i the corresponding sstem (1) has a unique singular point in the first quadrant and all trajectories of this quadrant tend to this point or to a limit ccle surrounding it when t. Proof. 1. The singular points of sstem (1) ' k k 3 0 ' k 1 k 3 0 A k 4 k 1 k k 1 4 k 1 k k k 3 k 1 k 3 k 1 4 k 1 k B k 4 k 1 k k 1 4 k 1 k k k 1 k 3 k 3 k 1 4 k 1 k A is located in the first quadrant and B is in the second one so onl the point A is of interest for us. The vector field on the coordinate aes bounding the first quadrant is directed inside the quadrant. If 0 : ' k 3 0 ' k 1 k 3 0 0 0 If 0 : ' k 0 ' k 1 0 Thus to understand the behavior of the trajectories in the quadrant we have to stud the singular points of the sstem at infinit.. Behavior of the trajectories at the ends of the O ais. a) Projection to the Poincaré sphere substitution and time rescaling: u z 1 dτ 1 z dt u' u u z u z k u z k 3 u z k 3 u z k 1 z 3 Uu z z' u z z k z 3 k 3 u z 3 Zu z () Singular points at the equator z 0: C0 0 and D1 0 but onl C corresponds to the first quadrant. C is degenerate blow-up First degree approimations of U and Z: U 1 u z Z 1 0 Characteristic function: F u Z 1 z U 1 zu z F 0 when z 0 or u z trajectories of () tend to C tangentiall to the lines z 0 and u z. b) Blow-up of the singular point 0 0 in () substitution: X u Y z u X ' X 1 X Y X Y k X Y k 3 X Y k 3 X 3 Y k 1 X Y 3 Y ' Y 1 Y k 3 X Y k 1 X Y 3 (3)

018-09-7.nb 11 Singular points at the ais X 0: 0 0 and 0 1 where 0 0 is a saddle. We investigate 0 1 further.. c) Moving the singular point 0 1 to 0 0 using the substitution w X v Y 1 and time rescaling dt dt w ' v w w v w k 1 w 3 k w 3 k 3 w 3 3 k 1 v w 3 k v w 3 k 3 v w 3 3 k 1 v w 3 k v w 3 k 3 v w 3 k 1 v 3 w 3 k 3 w 4 k 3 v w 4 k 3 v w 4 v ' v v k 1 w k 3 w 4 k 1 v w 3 k 3 v w 6 k 1 v w 3 k 3 v w 4 k 1 v 3 w k 3 v 3 w k 1 v 4 w (4) Application of a theorem b Andronov at al. (1973): Let 0 0 be an isolated equilibrium state of the sstem w ' P w v Pw v v ' v Q w v Qw v. Let v Φw be a solution of the equation v Q w v 0 in the neighborhood of 0 0 and assume that P w Φw w h.o.t. Then the origin is a saddle-node. (a) Phase portrait of sstem (4) (b) Phase portrait of sstem () after the blow-down (the direction of the trajectories changes since we divided b 1) Phase portraits of sstem (4) for fied values of k 1 and k 3 when k 1. 3. Behavior of the trajectories at the ends of the O ais 3. a) Projection to the Poincaré sphere substitution and time rescaling: u z 1 dτ 1 z dt

1 018-09-7.nb u' u u 3 u z u 3 z k 3 z k u z k 3 u z k 1 u z 3 Uu z z' u z u z k 3 z 3 k 1 z 4 Zu z (5) The origin is a degenerate singular point at the ends of O ais blow-up Second degree approimations of U and Z: U u k 3 z Z 0 Characteristic function: F u Z z U zu k 3 z F 0 when z 0 the characteristic direction is z 0 and a u-directional blow-up needs to be done 3. b) Blow-up in the u direction substitution and time rescaling: u X z X Y dt X dt and dividing b the common factor X X ' X 1 X X Y X Y k 3 Y k X Y k 3 X Y k 1 X Y 3 Y ' Y 1 X Y k 3 Y k X Y Phase portrait of sstem (6) and phase portrait of sstem (5) after the blowdown: (6) 4. Phase portrait of sstem (1) on the Poincaré disk There is eactl one singular point the point A in the first quadrant. Thus when the time increases each trajector of the first quadrant either reach the singular point A or a limit ccle surrounding A. 3.3 The big and the small limit ccle To simplif the further analsis we assume that the singular point A 0 0 of sstem (1) is on the straight line 1. Then ' k k 3 0 ' k 1 1 k 3 0 k 1 k 1 0 1 0 k 1 k 3

018-09-7.nb 13 Theorem. If In sstem (1) all parameters are positive k 3 1 k 1 k 1 k k 3 6 k 3 1 and k 3 1 g 1 1 43 k 3 9 k 3 3 k 3 3 1 k 3 3 k 3 3 k 3 k 3 0 then sstem (1) has a stable limit ccle and an additional unstable limit ccle bifurcates from the singular point A after small perturbations of the parameters. Proof. a) The singular point A 1 k 1 1 ' 1 k 3 1 k 3 of (1) is shifted into 0 0: k 1 4 k 3 1 k k 3 1 k 1 k 3 1 1 k 3 1 k 3 1 1 1 k 3 1 1 1 1 k 3 1 1 1 ' 1 (7) 1 k 3 1 k 1 k 3 1 1 k 1 k 3 1 1 k 3 1 k 3 1 1 1 k 3 1 1 1 1 k 3 1 1 Sstem (7) has onl one singular point in the first quadrant: 0 0. b) The Jacobian at the origin The trace of the Jacobian matri of (7) at the origin is tr 1 k 6 k 3 k k 3 k 3 tr 0 k 1 6 k 3 k 3 1 k 3 1 k 3 The eigenvalues are pure imaginar Λ 1 ±i Β where Β 1 k 3 3 k 3 1 k 3 if k 3 1. c) The matri S that transforms the Jacobian at the origin into Jordan canonical form: S 1 k 3 1 k 3 3 k 3 1 k 3 4 1 k 3 1 k 3 0 In (7) we introduce the change of coordinates 1 1 k 3 1 k 3 u 3 k 3 v 1 k 3 1 4 1 k 3 u 1 k 3 and time rescaling: dt dt 1 k 3 3 k 3 d) The transformed sstem (8): ' 1 k 3

14 018-09-7.nb u' 1 3 k 3 k 3 3 u 4 k 3 u k 3 u u 3 k 3 u 3 3 k 3 k 3 v 3 k 3 k 3 u v k 3 3 k 3 k 3 u v 4 3 k 3 k 3 u v 4 k 3 3 k 3 k 3 u v 3 v 4 k 3 v k 3 v 6 u v 8 k 3 u v k 3 u v 1 v ' 1 k 3 3 k 3 6 u 4 k 3 u k 3 u 11 u 13 k 3 u k 3 u k 3 3 u 6 u 3 k 3 u 3 6 k 3 u 3 k 3 3 u 3 10 3 k 3 k 3 u v 8 k 3 3 k 3 k 3 u v k 3 3 k 3 k 3 u v 1 3 k 3 k 3 u v 16 k 3 3 k 3 k 3 u v 4 k 3 3 k 3 k 3 u v 3 v 19 k 3 v 9 k 3 v k 3 3 v 18 u v 30 k 3 u v 14 k 3 u v k 3 3 u v The origin in (8) is either a center or a focus. To distinguish between the two cases we look for a Lapunov function. e) Lapunov s theorem. Let Φ be of the form Φu v u v and quantities g i satisfing the identit Φ u km3 Φ k m u k v m u' Φ v v ' g 1u v g u v 3... B comparing the coefficients of the corresponding powers on both sides: g 1 1 43 k 3 9 k 3 3 k 3 3 1 k 3 3 k 3 3 k 3 k 3 Remark: If Φ u u' Φ v v ' g 1 u 4 g u 6... then g 1 is a constant multiple of the previous result. When g 1 0 then Φu v is a positivel defined Lapunov function whose derivative is also positivel defined. Lapunov instabilit theorem the point A is an unstable focus Theorem 1 there is at least one stable limit ccle surrounding the point A From the trace tr 1 k 6 k 3 k k 3 k 3 we can see that we can slightl perturb the parameter k in 1 k 3 such a wa that the stabilit of the singular point A is changed. Therefore an unstable limit ccle appears near A as the result of the Andronov-Hopf bifurcation. Since the stable limit ccle is preserved after small perturbations the perturbed sstem has at least two limit ccles.

018-09-7.nb 15 3.4 Plotting the limit ccles Quit SetOptions AesStle ArrowheadsAutomatic & Plot ParametricPlot ListPlot ListLinePlot; SetDirectorNotebookDirector; SetOptions AesStle ArrowheadsAutomatic & Plot ListPlot ListLinePlot ListLogLogPlot ParametricPlot DateListPlot DiscretePlot; ClearAllk p q g; k 3 3 ; 0 k3 1 10 k 4 1; k 5 1; k k 3 4 k 3 k 4 k 3 k 5 k 5 1 k 3 k 5 10 000 ; k 1 k k 4 ; p : k k 4 k 3 k 5 ; q : k 1 k 4 k 3 k 5 ; ClearAllnsol ev plotter; nsol FirstNSolveJoin Thread p q 0 0 0; ev EigenvaluesDp q. nsol; plotterτ shift ag: Automatic pg: Automatic pp: 1000 ar: Automatic opts : Modulestartingpoint ss solution startingpoint. nsol shift; ss : NDSolveValueJoinu't put vt v't qut vt Threadu0 v0 startingpoint u v t Τ AccuracGoal ag PrecisionGoal pg opts; 1 trafopoint : point startingpoint; shift solutiont : trafothroughsst; ParametricPlotEvaluatesolutiont t 0 Τ Epilog Red PointSize0.05 Point0 0 Pointtrafo. nsol PlotRange All PlotPoints pp AspectRatio ar AesLabel LabelStle Directive14 ImageSize 50 PlotEvaluatesolutiont 1 t 0 Τ PlotRange All PlotPoints pp AesLabel t LabelStle Directive1 ImageSize 50 PlotEvaluatesolutiont t 0 Τ PlotRange All PlotPoints pp AesLabel t LabelStle Directive1 ImageSize 50

16 018-09-7.nb Big ccle trajector going inward. Distance from the singular point: 1 Figure1 plotter00 0 1 0.4 0. 0. 0.4 0.5 1.0 1.5.0 0.4 0. 0. 0.4 50 100 150 0.5 1.0 t 1.5.0 50 100 150 t

018-09-7.nb 17 Big ccle trajector going outward. Distance from the singular point: 10 5 Figure plotter30 0 0.00001 0.5 0.6 0.4 0. 0. 0.4 0.6 0.5 1.0 1.5.0.5 0.4 0. 0. 0.4 0.6 5 10 15 0 5 0.5 t 1.0 1.5.0.5 5 10 15 0 5 t

18 018-09-7.nb Small ccle trajector going inward. Distance from the singular point: 6 10 7 Figure3 plotter30 0 6. 10 7 0.4 0. 0. 0.4 0.5 1.0 1.5.0 0. 0. 5 10 15 0 5 0.5 t 1.0 1.5 5 10 15 0 5 t 0.4.0

018-09-7.nb 19 Small ccle trajector going inward and outward Figure41 plotter100 0 4. 10 1 13 100 10 000 1 1 1.0 0.5 0.5 1 3 0.5 0 40 60 80 t 1 0 40 60 80 t 0.5 1.0 3 Figure4 plotter100 0 4. 10 1 13 100 10 000 1 Method "BDF" 10 5 4 4 5 10 4 0 40 60 80 t 5 5 0 40 60 80 t 4 10

0 018-09-7.nb Figure43 plotter10 0 4. 10 1 13 100 10 000 1 Method "BDF" 0.5 0.5 1 3 0.5 4 6 8 t 1 4 6 8 t 0.5 3 Figure51 plotter10 0 10. 11 13 100 10 000 1 Method "BDF" 0.4 0. 0. 0.4 0.5 1.0 1.5.0 0. 0. 4 6 8 0.5 t 1.0 1.5 4 6 8 t 0.4.0

018-09-7.nb 1 Figure5 plotter100 0. 10 11 13 100 10 000 1 Method "BDF" 0.4 0. 0. 0.4 0.5 1.0 1.5.0 0. 0. 0 40 60 80 0.5 t 1.0 1.5 0 40 60 80 t 0.4.0 Figure53 plotter0 0 10. 1 13 100 10 000 1 Method "BDF" 4 1 1 4 6 8 1 1 5 10 15 4 t 4 6 8 5 10 15 t

018-09-7.nb 4 References [1] B. Ferčec I. Nag V. G. Romanovski G. Szederkéni and J. Tóth. Two Limit Ccles in a Two- Species Reaction. J. Math. Chem. (submitted) [] B. Ferčec I. Nag V. G. Romanovski G. Szederkéni and J. Tóth. Calculations for two limit ccles in a two-species reaction with wolfram mathematica (available 7 June 018). http://math.bme.hu/~nagi/mathematica_notebooks/inde.html [3] B. Ács G. Szederkéni Zs. Tuza and Z. A. Tuza. Computing all possible graph structures describing linearl conjugate realizations of kinetic sstems. Computer Phsics Communications 04:11-0 016. [4] A. A. Andronov E. A. Leontovich I. I. Gordon and A. G. Maier. Qualitative theor of second order differential equations. John Wile and Sons New York Toronto and Israel Program for Scientific Translations Jerusalem London 1973. [5] M. Dukarić. Qualitative studies of some polnomial sstems of ordinar differential equations. Doctoral dissertation 016 [6] F. Dumortier J. Llibre and J. Artes. Qualitative theor of planar differential sstems. Universitet. Springer-Verlag Berlin 006.