Qualitative studies of some biochemical models Valery Romanovski - PowerPoint PPT Presentation

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations By numerical studies Previte and Hoffman have shown that system (6) exhibits Hopf bifurcations, bistability, and chaotic phenomena. Al-khedhairi et al. have shown the existence of the transcritical, Hopf, Neimark-Staker and Fold Hopf bifurcations in system (6). Although both May-Leonard system (3) and system (6) are described by second degree polynomials and look similar, their dynamical behavior is very different: the dynamics of system (6) is much richer, than the one of (3). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations By numerical studies Previte and Hoffman have shown that system (6) exhibits Hopf bifurcations, bistability, and chaotic phenomena. Al-khedhairi et al. have shown the existence of the transcritical, Hopf, Neimark-Staker and Fold Hopf bifurcations in system (6). Although both May-Leonard system (3) and system (6) are described by second degree polynomials and look similar, their dynamical behavior is very different: the dynamics of system (6) is much richer, than the one of (3). In all models: Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations By numerical studies Previte and Hoffman have shown that system (6) exhibits Hopf bifurcations, bistability, and chaotic phenomena. Al-khedhairi et al. have shown the existence of the transcritical, Hopf, Neimark-Staker and Fold Hopf bifurcations in system (6). Although both May-Leonard system (3) and system (6) are described by second degree polynomials and look similar, their dynamical behavior is very different: the dynamics of system (6) is much richer, than the one of (3). In all models: Right hand sides are polynomial or rational functions Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations By numerical studies Previte and Hoffman have shown that system (6) exhibits Hopf bifurcations, bistability, and chaotic phenomena. Al-khedhairi et al. have shown the existence of the transcritical, Hopf, Neimark-Staker and Fold Hopf bifurcations in system (6). Although both May-Leonard system (3) and system (6) are described by second degree polynomials and look similar, their dynamical behavior is very different: the dynamics of system (6) is much richer, than the one of (3). In all models: Right hand sides are polynomial or rational functions Depend on many parameters Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations Elimination of variables How to eliminate some variables from the system: f 1 ( x 1 , . . . , x n ) = · · · = f m ( x 1 , . . . , x n ) = 0?? Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations Elimination of variables How to eliminate some variables from the system: f 1 ( x 1 , . . . , x n ) = · · · = f m ( x 1 , . . . , x n ) = 0?? Sylvester resultants Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations Elimination of variables How to eliminate some variables from the system: f 1 ( x 1 , . . . , x n ) = · · · = f m ( x 1 , . . . , x n ) = 0?? Sylvester resultants Gr¨ obner bases Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations Elimination of variables How to eliminate some variables from the system: f 1 ( x 1 , . . . , x n ) = · · · = f m ( x 1 , . . . , x n ) = 0?? Sylvester resultants Gr¨ obner bases The variety of the ideal I = � f 1 , . . . , f m � ⊂ k [ x 1 , . . . , x n ] in k n , denoted V ( I ), is the zero set of all polynomials of I , V ( I ) = { A = ( a 1 , . . . , a n ) ∈ k n | f ( A ) = 0 for all f ∈ I } , where k is a field, e.g. = Q , R C . We want to eliminate x 1 , . . . , x ℓ ( ℓ < n ) from f 1 ( x 1 , . . . , x n ) = · · · = f m ( x 1 , . . . , x n ) = 0. For an ideal I in k [ x 1 , . . . , x n ] we denote by V ( I ) its variety. Let us fix ℓ ∈ { 0 , 1 , . . . , n − 1 } . The ℓ -th elimination ideal of I is the ideal I ℓ = I ∩ k [ x ℓ +1 , . . . , x n ]. Any point ( a ℓ +1 , . . . , a n ) ∈ V ( I ℓ ) is called a partial solution of the system { f = 0 : f ∈ I } . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations The projection of a variety in k n onto k n − ℓ is not necessarily a variety. Theorem (Closure Theorem) Let V = V ( f 1 , . . . , f s ) be an affine variety in C n and let I ℓ be the ℓ -th elimination ideal for the ideal I = � f 1 , . . . , f s � . Then V ( I ℓ ) is the smallest affine variety containing π ℓ ( V ) ⊂ C n − ℓ (that is, V ( I ℓ ) is the Zariski closure of π ℓ ( V ) ). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations xy = 1 , xz = 1 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations xy = 1 , xz = 1 . Elimination ”by hand”: x = 1 / y , x = 1 / z , y � = 0 , z � = 0 = ⇒ x = 1 / a , y = a , z = a , a � = 0 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations xy = 1 , xz = 1 . Elimination ”by hand”: x = 1 / y , x = 1 / z , y � = 0 , z � = 0 = ⇒ x = 1 / a , y = a , z = a , a � = 0 . Elimination using the Elimination theorem: Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations xy = 1 , xz = 1 . Elimination ”by hand”: x = 1 / y , x = 1 / z , y � = 0 , z � = 0 = ⇒ x = 1 / a , y = a , z = a , a � = 0 . Elimination using the Elimination theorem: The reduced GB of I = � xy − 1 , xz − 1 � with lex x > y > z is { xz − 1 , y − z } . = ⇒ I 1 = � y − z � . = ⇒ V ( I 1 ) is the line y = z . Partial solutions are { ( a , a ) : a ∈ C } . ( a , a ) for which a � = 0 can be extended to (1 / a , a , a ), except of (0 , 0). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation A few examples of biochemical models Invariant surfaces in polynomial systems Basics of the elimination theory Limit cycle bifurcations xy = 1 , xz = 1 . Elimination ”by hand”: x = 1 / y , x = 1 / z , y � = 0 , z � = 0 = ⇒ x = 1 / a , y = a , z = a , a � = 0 . Elimination using the Elimination theorem: The reduced GB of I = � xy − 1 , xz − 1 � with lex x > y > z is { xz − 1 , y − z } . = ⇒ I 1 = � y − z � . = ⇒ V ( I 1 ) is the line y = z . Partial solutions are { ( a , a ) : a ∈ C } . ( a , a ) for which a � = 0 can be extended to (1 / a , a , a ), except of (0 , 0). Theorem (Extension Theorem) Let I = � f 1 , . . . , f s � be a nonzero ideal in the ring C [ x 1 , . . . , x n ] and let I 1 be the first elimination ideal for I. Write the generators of I in the form f j = g j ( x 2 , . . . , x n ) x N j 1 + ˜ g i , where N j ∈ { N ∪ 0 } , g j ∈ C [ x 2 , . . . , x n ] are nonzero polynomials, and ˜ g j are the sums of terms of f j of degree less than N j in x 1 . Consider a partial solution ( a 2 , . . . , a n ) ∈ V ( I 1 ) . If ( a 2 , . . . , a n ) �∈ V ( g 1 , . . . , g s ) , then there exists a 1 such that ( a 1 , a 2 , . . . , a n ) ∈ V ( I ) . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Conditions for existence of Hopf bifurcations x = x (1 − bx − y − z ) , ˙ ˙ y = y ( − c + x ) , ˙ z = z ( − e + fx + gy − β z ) . (6) System (6) has 6 equilibrium points, but all coordinates are positive only at A ( x 0 , y 0 , z 0 ), x 0 = c , y 0 = − b β c − β + cf − e , z 0 = c ( f − bg ) − e + g . (7) β + g bet + g The Jacobian at A is   − bc − c − c   − bc β + β + e − cf 0 0 J =  . (8)  β + g f ( − e + g + c ( f − bg )) g ( − e + g + c ( f − bg )) β ( e − cf + bcg − g ) β + g β + g β + g Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations The eigenvalues of J are complicated. The characteristic polynomial of J : 1 β + g (( − β − g ) u 3 +( β ( e − cf − g )+ bc ( β ( − 1+ g ) − g )) u 2 + p ( u ) = ( c ( e ( − 1+ f )+ f ( c − cf − g + bcg )+ β ( − 1+ b ( c + e − cf − g )+ b 2 cg ))) u − c ( β ( bc − 1) − e + cf )( e − cf − g + bcg ) . (9) Let u 1 = − b 0 be a real root of p ( u ). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations The eigenvalues of J are complicated. The characteristic polynomial of J : 1 β + g (( − β − g ) u 3 +( β ( e − cf − g )+ bc ( β ( − 1+ g ) − g )) u 2 + p ( u ) = ( c ( e ( − 1+ f )+ f ( c − cf − g + bcg )+ β ( − 1+ b ( c + e − cf − g )+ b 2 cg ))) u − c ( β ( bc − 1) − e + cf )( e − cf − g + bcg ) . (9) Let u 1 = − b 0 be a real root of p ( u ). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations The eigenvalues of J are complicated. The characteristic polynomial of J : 1 β + g (( − β − g ) u 3 +( β ( e − cf − g )+ bc ( β ( − 1+ g ) − g )) u 2 + p ( u ) = ( c ( e ( − 1+ f )+ f ( c − cf − g + bcg )+ β ( − 1+ b ( c + e − cf − g )+ b 2 cg ))) u − c ( β ( bc − 1) − e + cf )( e − cf − g + bcg ) . (9) Let u 1 = − b 0 be a real root of p ( u ). Thus, p ( u ) can be written in the form p ( u ) = − ( u + b 0 )( u 2 + w 2 ) ˜ (10) if two eigenvalues of J are pure imaginary ( u 1 , 2 = ± iw ). Equating the coefficients of u on both sides of p ( u ) = ˜ p ( u ): Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations bc ( β ( − g ) + β + g ) + b 0 ( β + g ) + β ( cf − e + g ) = 0 , � � bc 3 ( bg − f ) + c 2 ( be − 2 bg + f ) + b 0 w 2 + c ( g − e ) β + c 3 f ( bg − f ) + c 2 ( − beg + 2 ef − fg ) + b 0 gw 2 + ce ( g − e ) = 0 , (11) � bc 2 ( bg − f + 1) + c ( be − bg − 1) + w 2 � β + c 2 f ( bg − f + 1) + gw 2 + c ( e ( f − 1) − fg ) = 0 . p ( u ) = − ( u + b 0 )( u 2 + w 2 ) only for those p ( u ) can be represented as ˜ values of parameters of (6) for which system (11) has a solution. To find such values of parameters we eliminate from (9) b 0 , w . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations bc ( β ( − g ) + β + g ) + b 0 ( β + g ) + β ( cf − e + g ) = 0 , � � bc 3 ( bg − f ) + c 2 ( be − 2 bg + f ) + b 0 w 2 + c ( g − e ) β + c 3 f ( bg − f ) + c 2 ( − beg + 2 ef − fg ) + b 0 gw 2 + ce ( g − e ) = 0 , (11) � bc 2 ( bg − f + 1) + c ( be − bg − 1) + w 2 � β + c 2 f ( bg − f + 1) + gw 2 + c ( e ( f − 1) − fg ) = 0 . p ( u ) = − ( u + b 0 )( u 2 + w 2 ) only for those p ( u ) can be represented as ˜ values of parameters of (6) for which system (11) has a solution. To find such values of parameters we eliminate from (9) b 0 , w . • w should be different from zero. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations bc ( β ( − g ) + β + g ) + b 0 ( β + g ) + β ( cf − e + g ) = 0 , � � bc 3 ( bg − f ) + c 2 ( be − 2 bg + f ) + b 0 w 2 + c ( g − e ) β + c 3 f ( bg − f ) + c 2 ( − beg + 2 ef − fg ) + b 0 gw 2 + ce ( g − e ) = 0 , (11) � bc 2 ( bg − f + 1) + c ( be − bg − 1) + w 2 � β + c 2 f ( bg − f + 1) + gw 2 + c ( e ( f − 1) − fg ) = 0 . p ( u ) = − ( u + b 0 )( u 2 + w 2 ) only for those p ( u ) can be represented as ˜ values of parameters of (6) for which system (11) has a solution. To find such values of parameters we eliminate from (9) b 0 , w . • w should be different from zero. We add to (11) the equation 1 − vw = 0 , where v is a new variable, and then eliminate b 0 , w , v . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations obner basis ˜ We compute in Q [ v , w , b 0 , b , f , g , β, e , c ] a Gr¨ G (consists of 30 polynomials) of the ideal with respect to the lexicographic term order with v ≻ w ≻ b 0 ≻ b ≻ f ≻ g ≻ β ≻ e ≻ c and find that the third elimination ideal is � F � generated by F = b 3 β c 2 ( β ( − 1+ g ) − g ) g +( e − cf − g )( β f ( e − cf )+( β + e − ( β + c ) f ) g )+ b ( cg ( e (1 − f + g )+ f ( c ( − 1+ f − g )+ g ))+ β 2 ( c 2 f 2 +( e − g ) 2 + c (1 − 2 ef +2 fg ))+ β c ( cf ( − 1+ f + g − 2 fg )+ g (1+ f +2 g − 2 fg )+ e (1 − g + f ( − 1+2 g )))) − ( b 2 c ( cfg 2 + β 2 ( c + e − cf − g − 2 eg +2 cfg +2 g 2 )+ β g ( e − g + c (1+ g − fg )))) . (12) Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Denote by D the discriminant p ( u ). Theorem If all the coefficients of (6) and the coordinates of A are positive, then J has a pair of pure imaginary eigenvalues if and only if F = 0 and D < 0 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Denote by D the discriminant p ( u ). Theorem If all the coefficients of (6) and the coordinates of A are positive, then J has a pair of pure imaginary eigenvalues if and only if F = 0 and D < 0 . Proof. By the Closure Theorem for “almost all“ values of parameters b , f , g , β, e , c satisfying the condition F ( b , f , g , β, e , c ) = 0 our system has a solution. However it can happen that for some values of parameters it does not hold. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Denote by D the discriminant p ( u ). Theorem If all the coefficients of (6) and the coordinates of A are positive, then J has a pair of pure imaginary eigenvalues if and only if F = 0 and D < 0 . Proof. By the Closure Theorem for “almost all“ values of parameters b , f , g , β, e , c satisfying the condition F ( b , f , g , β, e , c ) = 0 our system has a solution. However it can happen that for some values of parameters it does not hold. We show that under the conditions of the theorem every solution of F ( b , f , g , β, e , c ) = 0 can be extended to a complete solution. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations obner basis ˜ The Gr¨ G contains the polynomials g 1 = ( g + β ) b 0 − bg β c + bgc + b β c + f β c + g β − β e ˜ g 2 = ( g + β ) w 2 + b 2 g β c 2 + bfgc 2 − bf β c 2 − bg β c + b β ec + ˜ b β c 2 − f 2 c 2 − fgc + fec + fc 2 − β c − ec , g 3 = c ( − β c + b β c 2 + β e − ce + e 2 − β cf + c 2 f − cef − β g + b β cg − eg + ˜ bceg ) v + β cw + b β cw − β ew + β cfw + β gw + cgw + bcgw − b β cgw g 4 = ( c ( β + e )( − e + cf ) 2 ( β + g ) 2 ) v + h 4 ( β, c , e , f , g , b , w ), ˜ where h 4 has a long expression. Theorem (Extension Theorem) Let I = � f 1 , . . . , f s � be a nonzero ideal in the ring C [ x 1 , . . . , x n ] and let I 1 be the first elimination ideal for I. Write the generators of I in the form f j = g j ( x 2 , . . . , x n ) x N j 1 + ˜ g i , where N j ∈ { N ∪ 0 } , g j ∈ C [ x 2 , . . . , x n ] are nonzero polynomials, and ˜ g j are the sums of terms of f j of degree less than N j in x 1 . Consider a partial solution ( a 2 , . . . , a n ) ∈ V ( I 1 ) . If ( a 2 , . . . , a n ) �∈ V ( g 1 , . . . , g s ) , then there exists a 1 such that ( a 1 , a 2 , . . . , a n ) ∈ V ( I ) . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations The coefficient of b 0 in ˜ g 1 does not vanish for the positive values of parameters, by the Extension Theorem (ET) every positive solution (ˆ b , ˆ g , ˆ f , ˆ β, ˆ e , ˆ c ) of F = 0 can be extended to (ˆ b 0 , ˆ b , ˆ g , ˆ f , ˆ β, ˆ e , ˆ c ) in the variety of J 2 . From the form of ˜ g 1 ⇒ ˆ = b 0 is real. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations The coefficient of b 0 in ˜ g 1 does not vanish for the positive values of parameters, by the Extension Theorem (ET) every positive solution (ˆ b , ˆ g , ˆ f , ˆ β, ˆ e , ˆ c ) of F = 0 can be extended to (ˆ b 0 , ˆ b , ˆ g , ˆ f , ˆ β, ˆ e , ˆ c ) in the variety of J 2 . From the form of ˜ g 1 ⇒ ˆ = b 0 is real. ⇒ the partial solution (ˆ b 0 , ˆ b , ˆ g , ˆ (˜ g 2 and the ET) = f , ˆ β, ˆ e , ˆ c ) w , ˆ b 0 , ˆ b , ˆ g , ˆ can be extended to a point ( ˆ f , ˆ β, ˆ e , ˆ c ) in the variety of J 1 . g 3 = c ( − β c + b β c 2 + β e − ce + e 2 − β cf + c 2 f − cef − β g + b β cg − ˜ eg + bceg ) v + β cw + b β cw − β ew + β cfw + β gw + cgw + bcgw − b β cgw g 4 = ( c ( β + e )( − e + cf ) 2 ( β + g ) 2 ) v + h 4 ( β, c , e , f , g , b , w ). ˜ w , ˆ b 0 , ˆ b , ˆ g , ˆ (˜ g 3 , ˜ g 4 and the ET) = ⇒ the partial solution ( ˆ f , ˆ β, ˆ e , ˆ c ) can be extended to a complete solution unless e − cf = bc − 1 = 0 . However in such case A has coordinates ( c , 0 , 0), which contradicts our assumption that all coordinates of A are positive. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Thus, if the parameters of (6) satisfy F = 0, then p ( u ) = − ( u + b 0 )( u 2 + w 2 ) ˜ Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Thus, if the parameters of (6) satisfy F = 0, then p ( u ) = − ( u + b 0 )( u 2 + w 2 ) ˜ w can be complex (pure imaginary)! Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Thus, if the parameters of (6) satisfy F = 0, then p ( u ) = − ( u + b 0 )( u 2 + w 2 ) ˜ w can be complex (pure imaginary)! A cubic polynomial with real coefficients has a pair of complex conjugate roots if and only if its discriminant is negative. Since D < 0 if the roots are α ± i γ , p ( u ) = ( u + b 0 )( u 2 − 2 α u + α 2 + γ 2 ) we conclude p ( u ) has two complex roots if w = γ is real, in which case the roots are u 1 , 2 = ± iw . � Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Thus, if the parameters of (6) satisfy F = 0, then p ( u ) = − ( u + b 0 )( u 2 + w 2 ) ˜ w can be complex (pure imaginary)! A cubic polynomial with real coefficients has a pair of complex conjugate roots if and only if its discriminant is negative. Since D < 0 if the roots are α ± i γ , p ( u ) = ( u + b 0 )( u 2 − 2 α u + α 2 + γ 2 ) we conclude p ( u ) has two complex roots if w = γ is real, in which case the roots are u 1 , 2 = ± iw . � g 2 = ( g + β ) w 2 + b 2 g β c 2 + bfgc 2 − bf β c 2 − bg β c + b β ec + Remark. ˜ b β c 2 − f 2 c 2 − fgc + fec + fc 2 − β c − ec . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Thus, if the parameters of (6) satisfy F = 0, then p ( u ) = − ( u + b 0 )( u 2 + w 2 ) ˜ w can be complex (pure imaginary)! A cubic polynomial with real coefficients has a pair of complex conjugate roots if and only if its discriminant is negative. Since D < 0 if the roots are α ± i γ , p ( u ) = ( u + b 0 )( u 2 − 2 α u + α 2 + γ 2 ) we conclude p ( u ) has two complex roots if w = γ is real, in which case the roots are u 1 , 2 = ± iw . � g 2 = ( g + β ) w 2 + b 2 g β c 2 + bfgc 2 − bf β c 2 − bg β c + b β ec + Remark. ˜ b β c 2 − f 2 c 2 − fgc + fec + fc 2 − β c − ec . Remark. Elimination ideals for studying such problem were used recently in N. Kruff, S. Walcher. Coordinate-independent criteria for Hopf bifurcations. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcdss.2020075 Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations The condition F = 0 , D < 0 is rather general. We can use Reduce . of Mathematica for some simplification. Example . In (6) let us set e = 5 , g = 3 , β = 2 and c = 4. Then Reduce[F == 0 && D < 0 && b >0 && f > 0 && y0>0 && z0 >0, {f, yields � √ � 1 2 < f < 1 46 − 2 4 and b is a root of the cubic equation, with respect to α , 21 − 50 f + 8 f 2 + 16 f 3 + (180 − 68 f − 24 f 2 ) α + ( − 168 − 88 f ) α 2 + 48 α 3 = 0 . If these conditions are fulfilled then the corresponding system (6) has a center manifold passing through the point A and the Jacobian at A has a pair of pure imaginary eigenvalues. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Remark. The approach can be used for studying similar problems. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Remark. The approach can be used for studying similar problems. x 1 = − w 1 x 2 + h . o . t . ˙ x 2 = w 1 x 1 + h . o . t . ˙ x 3 = − w 2 x 4 + h . o . t . ˙ x 2 = w 2 x 3 + h . o . t . ˙ ...................... x k +4 = ............... ˙ Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Remark. The approach can be used for studying similar problems. x 1 = − w 1 x 2 + h . o . t . ˙ x 2 = w 1 x 1 + h . o . t . ˙ x 3 = − w 2 x 4 + h . o . t . ˙ x 2 = w 2 x 3 + h . o . t . ˙ ...................... x k +4 = ............... ˙ Problem of existence of two integrals, bifurcation of invariant tori etc. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant surfaces in polynomial systems Limit cycle bifurcations Remark. The approach can be used for studying similar problems. x 1 = − w 1 x 2 + h . o . t . ˙ x 2 = w 1 x 1 + h . o . t . ˙ x 3 = − w 2 x 4 + h . o . t . ˙ x 2 = w 2 x 3 + h . o . t . ˙ ...................... x k +4 = ............... ˙ Problem of existence of two integrals, bifurcation of invariant tori etc. p ( u ) = a ( p k u k + · · · + p 1 u + p 0 )( u 2 + w 2 1 )( u 2 + w 2 ˜ 2 ) Eliminate p k , . . . , p 1 , p 0 , w 1 , w 2 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To understand the dynamics of a model described by systems of ODEs it is important to know: Singular points First integrals Invariant surfaces Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Invariant surfaces in polynomial systems x = P ( x , y , z ) , ˙ y = Q ( x , y , z ) , ˙ z = R ( x , y , z ) , ˙ (13) the maximal degree of polynomials P , Q , R is m . Definition A surface H = 0 ( H is a polynomial) is an invariant surface of (13) iff X ( H ) := ∂ H ∂ x P + ∂ H ∂ y Q + ∂ H ∂ z R = K H (14) K – a polynomial of degree at most m − 1. H – a Darboux polynomial of (13) K – a cofactor. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Invariant planes in May-Leonard system Problem: find all invariant planes of May-Leonard system x = x (1 − x − α 1 y − β 1 z ) , ˙ ˙ y = y (1 − β 2 x − y − α 2 z ) , ˙ z = z (1 − α 3 x − β 3 y − z ) . H ( x , y , z ) = h 000 + h 100 x + h 010 y + h 001 z . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Invariant planes in May-Leonard system Problem: find all invariant planes of May-Leonard system x = x (1 − x − α 1 y − β 1 z ) , ˙ ˙ y = y (1 − β 2 x − y − α 2 z ) , ˙ z = z (1 − α 3 x − β 3 y − z ) . H ( x , y , z ) = h 000 + h 100 x + h 010 y + h 001 z . Theorem System (3) has an invariant plane passing through the origin and different from the planes x = 0 , y = 0 , and z = 0 if one of the following conditions holds: 1) α 2 = β 1 , β 2 � = 1 , 2) α 1 = β 3 , α 3 � = 1 , 3) α 3 = β 2 , β 3 � = 1 , 4) β 3 = 2 − α 1 − α 2 + α 1 α 2 − α 3 + α 1 α 3 + α 2 α 3 − α 1 α 2 α 3 − β 1 − β 2 + β 1 β 2 , ( β 1 − 1)( β 2 − 1) 5) β 1 = α 3 = 1 , ( − 1 + α 1 )( − 1 + β 3 ) � = 0 , 6) β 2 = 1 , α 1 ( − 1 + α 2 )( − 1 + β 1 ) � = 0 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Proof. We look for an invariant plane in the form H ( x , y , z ) = h 100 x + h 010 y + h 001 z . (15) with the corresponding cofactor K ( x , y , z ) = c 0 + c 1 x + c 2 y + c 3 z . (16) Substituting H ( x , y , z ) and K ( x , y , z ) into X ( H ) = KH and comparing the coefficients of similar terms: g 1 = g 2 = · · · = g 9 = 0 (17) where g 1 = h 001 − c 0 h 001 , g 2 = − h 001 − c 3 h 001 , g 3 = h 010 − c 0 h 010 , g 4 = − h 010 − c 2 h 010 , g 5 = − β 3 h 001 − c 2 h 001 − α 2 h 010 − c 3 h 010 , (18) g 6 = h 100 − c 0 h 100 , g 7 = − h 100 − c 1 h 100 , g 8 = − β 2 h 010 − c 1 h 010 − α 1 h 100 − c 2 h 100 , g 9 = − α 3 h 001 − c 1 h 001 − β 1 h 100 − c 3 h 100 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations We are looking for planes passing through the origin ⇒ h 0 = 0. Denote by J = � g 1 , g 2 , . . . , g 9 � the ideal generated by polynomials of system (18). To obtain the conditions for existence of invariant planes we have to eliminate from (18) the variables h i and c i , that is, to compute the 7-th elimination ideal of J in the ring Q [ h , c , α, β ] := Q [ h 100 , h 010 , h 001 , c 0 , c 1 , c 2 , c 3 , α 1 , α 2 , α 3 , β 1 , β 2 , β 3 ] . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations X ( H ) = KH always has the solution H = 0, K = 0 and the solutions H 1 = x , H 2 = y , H 3 = z Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations X ( H ) = KH always has the solution H = 0, K = 0 and the solutions H 1 = x , H 2 = y , H 3 = z = ⇒ system (17) always has a solution Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations X ( H ) = KH always has the solution H = 0, K = 0 and the solutions H 1 = x , H 2 = y , H 3 = z = ⇒ system (17) always has a solution = ⇒ 7-th elimination ideal J is � 0 � . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations X ( H ) = KH always has the solution H = 0, K = 0 and the solutions H 1 = x , H 2 = y , H 3 = z = ⇒ system (17) always has a solution = ⇒ 7-th elimination ideal J is � 0 � . We impose the condition that polynomial (15) is not a constant and it is different from H 1 , H 2 , H 3 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations X ( H ) = KH always has the solution H = 0, K = 0 and the solutions H 1 = x , H 2 = y , H 3 = z = ⇒ system (17) always has a solution = ⇒ 7-th elimination ideal J is � 0 � . We impose the condition that polynomial (15) is not a constant and it is different from H 1 , H 2 , H 3 . H ( x , y , z ) = h 100 x + h 010 y + h 001 z defines a plane different from x = 0 , y = 0, z = 0 if at least two from the coefficients h 100 , h 010 , h 001 are different from zero. In the polynomial form: 1 − wh 100 h 010 = 0, 1 − wh 100 h 001 = 0, 1 − wh 010 h 001 = 0 with w being a new variable. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find systems admitting invariant surfaces with h 100 h 010 � = 0: Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find systems admitting invariant surfaces with h 100 h 010 � = 0: • compute (e.g. using the routine eliminate of Singular ) the 8-th elimination ideal of the ideal J (1) = � J , 1 − wh 100 h 010 � , in the ring Q [ w , h , c , α, β ] := Q [ w , h 100 , h 010 , h 001 , c 0 , c 1 , c 2 , c 3 , α 1 , α 2 , α 3 , β 1 , β 2 , β 3 ] Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find systems admitting invariant surfaces with h 100 h 010 � = 0: • compute (e.g. using the routine eliminate of Singular ) the 8-th elimination ideal of the ideal J (1) = � J , 1 − wh 100 h 010 � , in the ring Q [ w , h , c , α, β ] := Q [ w , h 100 , h 010 , h 001 , c 0 , c 1 , c 2 , c 3 , α 1 , α 2 , α 3 , β 1 , β 2 , β 3 ] Denote this elimination ideal by J (1) 7 ; its variety by V 1 = V ( J (1) 7 ). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find all the possibles invariant surfaces: Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find all the possibles invariant surfaces: • Proceeding analogously, find other two eliminations ideals J (2) 7 , J (3) 7 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find all the possibles invariant surfaces: • Proceeding analogously, find other two eliminations ideals J (2) 7 , J (3) 7 . Denote the corresponding varieties V 2 = V ( J (2) 7 ) , V 3 = V ( J (3) 7 ). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find all the possibles invariant surfaces: • Proceeding analogously, find other two eliminations ideals J (2) 7 , J (3) 7 . Denote the corresponding varieties V 2 = V ( J (2) 7 ) , V 3 = V ( J (3) 7 ). • The union, V = V 1 ∪ V 2 ∪ V 3 , contains the set of all May-Leonard asymmetric systems, (3), having invariant planes passing through the origin. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find all the possibles invariant surfaces: • Proceeding analogously, find other two eliminations ideals J (2) 7 , J (3) 7 . Denote the corresponding varieties V 2 = V ( J (2) 7 ) , V 3 = V ( J (3) 7 ). • The union, V = V 1 ∪ V 2 ∪ V 3 , contains the set of all May-Leonard asymmetric systems, (3), having invariant planes passing through the origin. • Since V = V ( J (1) 7 ) ∪ V ( J (2) 7 ) ∪ V ( J (3) 7 ) = V ( J (1) ∩ J (2) ∩ J (3) 7 ), 7 7 to find the irreducible decomposition of V : Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find all the possibles invariant surfaces: • Proceeding analogously, find other two eliminations ideals J (2) 7 , J (3) 7 . Denote the corresponding varieties V 2 = V ( J (2) 7 ) , V 3 = V ( J (3) 7 ). • The union, V = V 1 ∪ V 2 ∪ V 3 , contains the set of all May-Leonard asymmetric systems, (3), having invariant planes passing through the origin. • Since V = V ( J (1) 7 ) ∪ V ( J (2) 7 ) ∪ V ( J (3) 7 ) = V ( J (1) ∩ J (2) ∩ J (3) 7 ), 7 7 to find the irreducible decomposition of V : compute the ideal J = J (1) ∩ J (2) ∩ J (3) (routine intersect 7 7 7 of Singular ); find the irreducible decomposition of V ( J ) (routine minAssGTZ of Singular ). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations To find all the possibles invariant surfaces: • Proceeding analogously, find other two eliminations ideals J (2) 7 , J (3) 7 . Denote the corresponding varieties V 2 = V ( J (2) 7 ) , V 3 = V ( J (3) 7 ). • The union, V = V 1 ∪ V 2 ∪ V 3 , contains the set of all May-Leonard asymmetric systems, (3), having invariant planes passing through the origin. • Since V = V ( J (1) 7 ) ∪ V ( J (2) 7 ) ∪ V ( J (3) 7 ) = V ( J (1) ∩ J (2) ∩ J (3) 7 ), 7 7 to find the irreducible decomposition of V : compute the ideal J = J (1) ∩ J (2) ∩ J (3) (routine intersect 7 7 7 of Singular ); find the irreducible decomposition of V ( J ) (routine minAssGTZ of Singular ). The output gives the 6 conditions of the theorem. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Invariant surfaces of degree 2: H ( x , y , z ) = 1+ h 100 x + h 010 y + h 001 z + h 200 x 2 + h 110 xy + h 101 xz + h 020 y 2 + h 011 yz + h 002 z 2 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Invariant surfaces of degree 2: H ( x , y , z ) = 1+ h 100 x + h 010 y + h 001 z + h 200 x 2 + h 110 xy + h 101 xz + h 020 y 2 + h 011 yz + h 002 z 2 . The computational procedure yields 88 conditions on the parameters α i , β i of the May-Leonard asymmetric system for existence of an invariant surface of degree two not passing through the origin. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Invariant surfaces of degree 2: H ( x , y , z ) = 1+ h 100 x + h 010 y + h 001 z + h 200 x 2 + h 110 xy + h 101 xz + h 020 y 2 + h 011 yz + h 002 z 2 . The computational procedure yields 88 conditions on the parameters α i , β i of the May-Leonard asymmetric system for existence of an invariant surface of degree two not passing through the origin. We say, that two conditions for existence of invariant surfaces are conjugate if one can be obtained from another by means of one of transformations: α 1 → α 3 , β 1 → β 3 , α 2 → α 1 , β 2 → β 1 , α 3 → α 2 , β 3 → β 2 , α 1 → α 2 , β 1 → β 2 , α 2 → α 3 , β 2 → β 3 , α 3 → α 1 , β 3 → β 1 , α 1 → β 2 , β 1 → α 2 , α 2 → β 1 , β 2 → α 1 , α 3 → β 3 , β 3 → α 3 , α 1 → β 3 , β 1 → α 3 , α 2 → β 2 , β 2 → α 2 , α 3 → β 1 , β 3 → α 1 , α 1 → β 1 , β 1 → α 1 , α 2 → β 3 , β 2 → α 3 , α 3 → β 2 , β 3 → α 2 , Theorem System (3) has an irreducible invariant surface not passing through the origin if one of the following conditions or conjugated to it holds: Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations 1 α 2 = β 1 = β 2 − 1 / 2 = α 1 − 3 = 0 2 α 2 = β 1 = β 2 − 3 = α 1 − 3 = 0 β 3 = β 1 = α 3 + β 2 − 1 = α 2 + 1 = α 1 − α 3 − 1 = 0 3 4 β 3 = β 1 = α 3 + 1 = β 2 − 3 = α 2 + 1 = α 1 − 1 / 2 = 0 5 β 3 = β 1 = α 3 − 3 = β 2 − 3 = α 2 − 3 / 2 = α 1 + 1 = 0 6 β 1 = β 3 − 3 = α 3 − 3 = β 2 − 1 = α 2 − 1 / 2 = α 1 − 1 = 0 7 β 1 = β 3 − 3 = α 3 − 1 / 2 = β 2 − 1 / 2 = α 2 + 1 = α 1 − 3 = 0 8 β 1 = β 3 − 1 / 2 = α 3 − 3 = β 2 − 3 = α 2 − 3 / 2 = α 1 − 1 / 2 = 0 9 β 1 = β 3 − 3 = α 3 + 3 = β 2 − 3 = α 2 + 1 = α 1 − 3 = 0 10 β 1 = β 3 − 1 / 2 = α 3 − 2 = β 2 − 3 = α 2 − 3 / 2 = α 1 − 1 / 2 = 0 11 β 1 = α 3 = β 2 − β 3 − 1 = α 2 + β 3 − 2 = α 1 + β 3 − 1 = 0 12 β 1 = β 3 − 3 = α 3 + β 2 − 4 = α 2 + 1 = α 1 − α 3 + 2 = 0 13 β 3 − 1 / 2 = α 3 − 1 / 2 = α 2 − 3 = β 1 − 3 = α 1 + β 2 − 2 = 0 14 β 3 − 1 / 2 = α 3 − 3 = β 2 − 3 = α 2 − 3 = β 1 − 3 = α 1 − 1 / 2 = 0 15 β 3 − 1 / 2 = β 2 − 3 = α 2 − 3 = α 3 + β 1 − 2 = α 1 − 1 / 2 = 0 16 β 3 − 3 = α 3 − 3 = α 2 − 3 = β 1 − 3 = α 1 + β 2 − 2 = 0 17 β 3 − 3 = α 3 + β 2 − 4 = α 2 − 3 = α 3 + β 1 − 2 = α 1 − α 3 + 2 = 0 Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Modular computations Computational complexity of the Gr¨ obner basis calculations over the field of rational numbers is an essential obstacle for using the Gr¨ obner basis theory for the real world applications. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Modular computations Computational complexity of the Gr¨ obner basis calculations over the field of rational numbers is an essential obstacle for using the Gr¨ obner basis theory for the real world applications. For finding the surfaces of the second degree the computations over the field Z p were used. H ( x , y , z ) = 1 + h 100 x + h 010 y + h 001 z + h 200 x 2 + h 110 xy + h 101 xz + h 020 y 2 + h 011 yz + h 002 z 2 . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Modular computations Computational complexity of the Gr¨ obner basis calculations over the field of rational numbers is an essential obstacle for using the Gr¨ obner basis theory for the real world applications. For finding the surfaces of the second degree the computations over the field Z p were used. H ( x , y , z ) = 1 + h 100 x + h 010 y + h 001 z + h 200 x 2 + h 110 xy + h 101 xz + h 020 y 2 + h 011 yz + h 002 z 2 . Modular computations: Choose a prime number p and do all calculations modulo p , that is, in Z p = Z / p . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Reconstruct (lift) r / s ∈ Q given its image t ∈ Z p . Algorithm by P. Wang ( ⌊·⌋ stands for the floor function): Step 1. u = ( u 1 , u 2 , u 3 ) := (1 , 0 , m ) , v = ( v 1 , v 2 , v 3 ) := (1 , 0 , c ) � Step 2. While m / 2 ≤ v 3 do { q := ⌊ u 3 / v 3 ⌋ , r := u − qv , u := v , v := r } � Step 3. If | v 2 | ≥ m / 2 then error() Step 4. Return v 3 , v 2 Given an integer c and a prime number p the algorithm produces integers v 3 and v 2 such that v 3 / v 2 ≡ c ( mod p ), that is, v 3 = v 2 c + pt with some t . If such a number v 3 / v 2 does need not exist. If this is the case, then the algorithm returns ”error()”. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Example f 1 =8 x 2 y 2 + 5 xy 3 + 3 x 3 z + x 2 yz , f 2 = x 5 + 2 y 3 z 2 + 13 y 2 z 3 + 5 yz 4 , (19) f 3 =8 x 3 + 12 y 3 + xz 2 + 3 , f 4 =7 x 2 y 4 + 18 xy 3 z 2 + y 3 z 3 . Under the lexicographic ordering with x > y > z a Groebner basis for I is G = { x , y 3 + 1 4 , z 2 . } (20) Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Example f 1 =8 x 2 y 2 + 5 xy 3 + 3 x 3 z + x 2 yz , f 2 = x 5 + 2 y 3 z 2 + 13 y 2 z 3 + 5 yz 4 , (19) f 3 =8 x 3 + 12 y 3 + xz 2 + 3 , f 4 =7 x 2 y 4 + 18 xy 3 z 2 + y 3 z 3 . Under the lexicographic ordering with x > y > z a Groebner basis for I is G = { x , y 3 + 1 4 , z 2 . } (20) Computing in the field Z 32003 : G ′ = { x , y 3 + 8001 , z 2 . } (21) Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Example f 1 =8 x 2 y 2 + 5 xy 3 + 3 x 3 z + x 2 yz , f 2 = x 5 + 2 y 3 z 2 + 13 y 2 z 3 + 5 yz 4 , (19) f 3 =8 x 3 + 12 y 3 + xz 2 + 3 , f 4 =7 x 2 y 4 + 18 xy 3 z 2 + y 3 z 3 . Under the lexicographic ordering with x > y > z a Groebner basis for I is G = { x , y 3 + 1 4 , z 2 . } (20) Computing in the field Z 32003 : G ′ = { x , y 3 + 8001 , z 2 . } (21) Rational reconstruction yields (20). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Calculations for the case H ( x , y , z ) = h 100 x + h 010 y + h 001 z + h 200 x 2 + h 110 xy + h 101 xz + h 020 y 2 + h 011 yz turned out computationally unfeasible even over Z p . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Darboux first integral Let n be an arbitrary natural number, H i be algebraic invariant surfaces of x = P ( x , y , z ) , ˙ y = Q ( x , y , z ) , ˙ z = R ( x , y , z ) , ˙ (22) with the corresponding cofactors K i ( i = 1 , 2 , . . . , n ). A Darboux first integral of system (22) is a function of the form n � H i ( x , y , z ) λ i , Ψ( x , y , z ) = i =1 where n � λ i K i = 0 (23) i =1 and λ i are some constants. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Darboux first integral Let n be an arbitrary natural number, H i be algebraic invariant surfaces of x = P ( x , y , z ) , ˙ y = Q ( x , y , z ) , ˙ z = R ( x , y , z ) , ˙ (22) with the corresponding cofactors K i ( i = 1 , 2 , . . . , n ). A Darboux first integral of system (22) is a function of the form n � H i ( x , y , z ) λ i , Ψ( x , y , z ) = i =1 where n � λ i K i = 0 (23) i =1 and λ i are some constants. Using the obtained invariant surface a number of Darboux first integrals of the May-Leonard system was constructed. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Periodic solutions in the May-Leonard system Chi, Hsu and Wu (SIAM J. Appl. Math. 1998) have shown that the ML system can have a family of periodic solutions Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Periodic solutions in the May-Leonard system Chi, Hsu and Wu (SIAM J. Appl. Math. 1998) have shown that the ML system can have a family of periodic solutions but they mentioned that the family arises as the result of Hopf bifurcations. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Periodic solutions in the May-Leonard system Chi, Hsu and Wu (SIAM J. Appl. Math. 1998) have shown that the ML system can have a family of periodic solutions but they mentioned that the family arises as the result of Hopf bifurcations. In fact there is another mechanism for existence of the family. Under condition 4) of Theorem 2 we have: β 3 = 2 − α 1 − α 2 + α 1 α 2 − α 3 + α 1 α 3 + α 2 α 3 − α 1 α 2 α 3 − β 1 − β 2 + β 1 β 2 , ( β 1 − 1)( β 2 − 1) H 4 = − x + α 3 x + β 2 x − α 3 β 2 x + y − α 1 y − α 3 y + α 1 α 3 y + z − β 1 z − β 2 z + β 1 β 2 z (24) x = 0 , y = 0 , z = 0 Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Periodic solutions in the May-Leonard system Chi, Hsu and Wu (SIAM J. Appl. Math. 1998) have shown that the ML system can have a family of periodic solutions but they mentioned that the family arises as the result of Hopf bifurcations. In fact there is another mechanism for existence of the family. Under condition 4) of Theorem 2 we have: β 3 = 2 − α 1 − α 2 + α 1 α 2 − α 3 + α 1 α 3 + α 2 α 3 − α 1 α 2 α 3 − β 1 − β 2 + β 1 β 2 , ( β 1 − 1)( β 2 − 1) H 4 = − x + α 3 x + β 2 x − α 3 β 2 x + y − α 1 y − α 3 y + α 1 α 3 y + z − β 1 z − β 2 z + β 1 β 2 z (24) x = 0 , y = 0 , z = 0 The Darboux first integral Ψ = x α 1 y α 2 z α 3 H α 4 (25) 4 α 1 ( − 1 + β 1 ) α 1 ( − 1 + β 1 )( − 1 + β 2 ) α 1 (1 − α 2 + α 2 α 3 − α 3 β 1 − β 2 + β 1 β 2 ) α 2 = − , α 3 = , α 4 = − . α 2 − 1 ( − 1 + α 2 )( − 1 + α 3 ) Valery Romanovski Qualitative studies of some biochemical models ( − 1 + α 2 )( − 1 + α 3 )

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations For simplicity we take the parameters β 1 = 1 / 4 , β 2 = 11 / 10 , α 1 = 5 / 4 , α 2 = 4 / 5 , α 3 = 3 / 2 , β 3 = 2 / 3. In this case system (3) x = x ( − x − 5 y 4 − z y = y ( − 11 x 10 − y − 4 z z = z (3 x 2 +2 y ˙ 4+1) , ˙ 5 +1) , ˙ 3 + z − 1) . (26) and the singular point P has the coordinates x 0 = 1 / 3 , y 0 = 1 / 2 , z 0 = 1 / 6 . Proposition System (26) has a family of periodic solutions in a neighborhood of the singular point P (1 / 3 , 1 / 2 , 1 / 6). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Proof: Moving the origin to the singular point by the substitution u = x − x 0 , v = y − y 0 , w = z − z 0 and then performing the linear change of coordinates u =2 X + 370 Y / 249 , √ v =3 X − Y − 15 10 Z / 83 , √ w = X + 1 / 249( − 235 Y + 77 10 Z ) we obtain from (26) Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations √ X = − X − 6 X 2 + 10450 Y 2 − 10450 Z 2 268671 + 38048 10 YZ ˙ 268671 , 806013 � � 2 16979 5 YZ 5 XZ − 2090 Y 2 + 2090 Z 2 Z 2 ˙ Y = √ − 6 XY + 39923 + 39923 , 39923 3 10 � √ √ 10 Y 2 10 Z 2 Y 5 XY − 6 XZ + 19187 2 + 7730 YZ 119769 − 19187 ˙ √ Z = − − . 119769 119769 3 10 By the Center Manifold Theorem ∃ an analytic center manifold X = h ( Y , Z ) passing through X = Y = Z = 0. Expanding the first integral (25) into power series Ψ( X , Y , Z ) = Y 2 + Z 2 + h . o . t . ⇒ in a neighborhood of the origin there exists a family of periodic orbits formed by the intersection of the graphs of X = h ( Y , Z ) and Ψ = c (0 < c < c 0 ). Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Lyapunov functions on the center manifold x = A x + F ( x ) = G ( x ) , ˙ (27) x = ( x , y , z ), the matrix A has the eigenvalues λ 1 , λ 2 , λ 3 and λ 1 < 0, λ 2 = i ω , λ 3 = − i ω , F is a vector-function, which is analytic in a neighborhood of the origin and such that its series expansion starts from quadratic or higher terms, and G ( x ) = ( G 1 ( x ) , G 2 ( x ) , G 3 ( x )) T . By the Center Manifold Theorem the system has a center manifold defined by a function x = f ( y , z ). After a linear transformation and rescaling of time system: − v + P ( u , v , w ) = � u = ˙ P ( u , v , w ) u + Q ( u , v , w ) = � (28) v = ˙ Q ( u , v , s ) w = − λ w + R ( u , v , w ) = � ˙ R ( u , v , w ) . Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Theorem Suppose that for (27) there exists a function � ∞ a klm x k y l z m Ψ( x ) = (29) k + l + m =2 X (Ψ) := ∂ Ψ( x ) ∂ x G 1 ( x ) + ∂ Ψ( x ) ∂ y G 2 ( x ) + ∂ Ψ( x ) ∂ z G 3 = g 1 ( y 2 + z 2 ) 2 + g 2 ( y 2 + z 2 ) 3 + . . . . (30) Let x = f ( y , z , α ∗ ) (31) be the center manifold of system (27) corresponding to the value α ∗ of parameters of the system and � a klm x k y l z m q ( x , α ∗ ) = (32) k + l + m =2 Valery Romanovski Qualitative studies of some biochemical models be the quadratic part of (29).

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Let q 1 ( y , z , α ∗ ) be q ( x , α ∗ ) evaluated on (31). Assume that q 1 ( y , z , α ∗ ) is positively defined quadratic form and g 1 ( α ∗ ) = g 2 ( α ∗ ) = · · · = g k ( α ∗ ) = 0 , g k +1 ( α ∗ ) � = 0 . (33) Then, 1) if g k +1 ( α ∗ ) < 0, the corresponding system (27) has a stable focus at the origin on the center manifold, and if g k +1 ( α ∗ ) > 0 then the focus is unstable. 2) if it is possible to choose perturbations of the parameters α in system (27) such that | g 1 ( α k ) | ≪ | g 2 ( α k − 1 ) | ≪ . . . | g k ( α 1 ) | ≪ | g k +1 ( α ∗ ) | , (34) α j +1 is arbitrary close to α j and the signs of g s ( α m ) in (34) alternate, then system (27) corresponding to the parameter α k has at least k limit cycles on the center manifold. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations Proof. 1) Since q 1 is positively defined the function Ψ restricted to the center manifold is positively defined in a small neighborhood of the origin. The derivative of Ψ with respect to the vector field on the center manifold has the same sign as g k +1 ( α ∗ ). Thus, by the Lyapunov theorem the origin is a stable focus on the center manifold if g k +1 ( α ∗ ) < 0 and unstable focus if g k +1 ( α ∗ ) > 0. Valery Romanovski Qualitative studies of some biochemical models

Introduction Necessary conditions for Hopf bifurcation Invariant planes in May-Leonard system Invariant surfaces in polynomial systems Invariant surfaces of degree 2 in May-Leonard system Limit cycle bifurcations 2) Assume for determinacy that g k +1 ( α ∗ ) < 0. Under the condition of the theorem the equality Ψ( x , α ∗ ) = c ( c ∈ (0 , c 1 ]) defines in a small neighborhood of the origin near the center manifold (31) a family of cylinders which are transversal to the center manifold. Let C 1 be the curve formed by the intersection of the cylinder Ψ( x , α ∗ ) = c 1 and the center manifold M ( α ∗ ) of system (27) defined by (31). If c 1 is sufficiently small then C 1 is an oval on M ( α ∗ ) and the vector field is directed inside C 1 , since X (Ψ( x , α ∗ )) = g k +1 ( α ∗ )( y 2 + z 2 ) k +2 + h . o . t and g k +1 ( α ∗ ) < 0 . Valery Romanovski Qualitative studies of some biochemical models