aa r X i v : . [ m a t h . D S ] S e p THE PITCHFORK BIFURCATION
INDIKA RAJAPAKSE AND STEVE SMALE
Abstract.
We give development of a new theory of the Pitchfork bifurcation,which removes the perspective of the third derivative and a requirement ofsymmetry.
Contents
1. Introduction2. Normal form3. A relationship of our normal form to the main biological example4. General pitchfork theorem5. A biological perspective1.
Introduction
The normal form for the pitchfork bifurcation is described usually for one variable(see Guckenheimer and Holmes) by(1) dxdt = F ( x ) = µx − x , x ∈ R , µ ∈ R . Note that this equation is invariant under the change of the variable x → − x. Thatis, F is an odd function. This condition suggests that the Pitchfork bifurcation isgeneric for problems that have symmetry. To obtain the above form one argues (orassumes) that the second derivative of F ( x ) is zero. This normal form is standardin the literature on Pitchfork bifurcation [2, 3].Our own proof [4] for the one variable case is different from the previous literature inthat a new uniformity condition is satisfied in place of symmetry or the hypothesis,vanishing of a second derivative. For the new uniformity condition see below. Inaddition we take the path of using this uniformity together with the Poincar´e HopfTheorem to show that the second derivative must be zero.We have found little in the literature on the case of more than one variable, exceptfor suggesting that (1) still applies. Kuznetsov [5] has a proof for an n variable casefor pitchfork that assumes invariance under an involution. Kuznetsov’s hypothesiseliminates a second derivative.Our paper [4] states an n − dimensional version of the pitchfork theorem. Thatproof involves a reduction to one dimensional theory using center manifold theory.In this paper we give a proof, which gives new insight especially for more than onevariable. Date : 8-28-2016.We extend thanks to James Gimlett and Srikanta Kumar at Defense Advanced ResearchProjects Agency for support and encouragement.
Figure 1.
Geometry of isoclines for the normal form (3). Left:a single equilibrium at (1 ,
1) for a <
1; middle: a single equi-librium at (1 ,
1) for a = 1; right: for a >
1, (1 ,
1) becomesunstable, and two stable equilibria emerge flanking (1 , a
2. Normal form
We propose that the normal form for the Pitchfork bifurcation for the dimensionof the space greater than one be: dxdt = y − ay − x (2) dydt = x − ax − y, x, y ≥ a, the ”central equilibrium” is ( x, y ) =(0 , , and may be described in terms of the two isoclines − the curves in the ( x, y )plane where dxdt = 0 (the x isoclines) and where dydt = 0 (the y isoclines). For given a the equilibria are given as the intersection of these isoclines. The isoclines aredescribed respectively, by the equations: x = y − ay (3a) y = x − ax. (3b)Figure 1 shows the curves for selected values of a. For a ≤ , the central equilibriumis the only equilibrium relevant for the bifurcation. For a > a = 1 , when the isoclines are tangent to each other at(0 , The defining characteristic of a pitchfork bifurcation is the transitionfrom a single stable equilibrium to two new stable equilibria separatedby a saddle. The saddle emerges from the old stable equilibrium.
HE PITCHFORK BIFURCATION 3
The Figure 1 illustrates this characteristic as a increases from less than one togreater than one.Next we will obtain the intersection points of two isoclines as in Equations 3a and3b. The curves described by Equations 3a and 3b are quadratic and one can solvetheir intersection points analytically. If we substitute 3b into 3a, solving for x yieldsfor the solutions x i as can be checked: x = 0 , central equilibrium(4a) x = 12 ( a −
1) + 12 p ( a −
1) ( a + 3) x = 12 ( a − − p ( a −
1) ( a + 3) x = a + 1Similarly, substituting Equation 3a into 3b and solving for y yields the solutions y i : y = 0 , central equilibrium(4b) y = 12 ( a − − p ( a −
1) ( a + 3) y = 12 ( a −
1) + 12 p ( a −
1) ( a + 3) y = a + 1 . The pairs ( x i , y i ) , i = 1 , .., x , y ) is extraneous to the bifurcation phenomena. Note also that the Equation4a and 4b show the bifurcation effect at a = 1 and a > . The two isoclines are parabolas and they get translated vertically and horizontallyas the parameter a increases. One can see the intersections of these parabolas interms of simple analytic geometry, and these intersections include the equilibria ofthe pitchfork . This formalism allows us to see the pitchfork variables in terms ofgeometry and extend the analysis to the nonsymmetric case, dxdt = y − ay − x, dydt = x − bx − y .The Jacobian matrix J of the first partial derivatives of System (2) at the centralequilibrium ( x , y ) = (0 ,
0) of Equation 3 is:(5) J = (cid:18) − − a − a − (cid:19) . For each a , the eigenvalues are λ = a − , λ = − a − − ,
1) and (1 ,
1) respectively. When a = 1 , λ = 0 and λ < . When a < , both λ , λ have negative real parts. Hence the central equilibrium isstable. When a > , λ > λ < , and the central equilibrium is a saddle.The qualitative structure is robust.Eigenvalues at the equilibrium ( x , y ) are given by: λ = − p − ( a − a + 3) + 3and λ = − − p − ( a − a + 3) + 3 . When 1 < a, both λ , λ have negative realparts. Hence the equilibrium is stable, similarly for ( x , y ). Remark 1 : When 1 . < a, both λ , λ are complex conjugate numbers withnegative real parts. The pitchfork phenomena continues after a = 1 . INDIKA RAJAPAKSE AND STEVE SMALE
3. A relationship of our normal form (2) to a main biological example
We are motivated by work by Gardner et al. [6] for a ”synthetic, bistable gene-regulatory network . . . [to] provide a simple theory that predicts the conditionsnecessary for bistability.” The toggle, as designed and constructed by Gardner etal., is a network of two mutually inhibitory genes that acts as a switch by somemechanism, as a control for switching from one basin to another. Consider theparticular setting of Gardner et al.’s circuit design of the toggle switch as: dxdt = α y m − x (6) dydt = α x n − y. If m = n = 0 , the equilibrium is x = α , y = α , and the eigenvalues of theJacobian are negative. If α < x ) and α < y ) , the system has aunique global stable equilibrium [6, 7].When α = α = 2 , and m = n > , the system in Equation 6 can be written morespecifically as in Gardner et al.: dxdt = 21 + y m − x = f ( x, y )(7) dydt = 21 + x m − y = g ( x, y ) , ≤ x, y. For this system with 0 ≤ m ≤ , every equilibria must be (1 , . We give theTaylor approximation for each m about the equilibrium (1 , f x , f y , f xx , f yy , f xy and similar for g , all at the point(1 , . These can be computed as: f x = − , f y = − m, g x = − m, g y = − f yy = g xx = 12 m, and the remaining derivatives are zero.Therefore, the Taylor approximation about the equilibrium (1 ,
1) for each m ≤ dxdt = 12 my − my + m + 1 − x (8a) dydt = 12 mx − mx + m + 1 − y (8b)Recall our normal form Equation 1 written in terms of u and v is: dudt = v − av − u (9a) dvdt = u − au − v. (9b)We wish to compare Equation 8 and 9 with the parameter values at the bifurcationpoints, namely m = 2 for Equation 8 and a = 1 for Equation 9. Then the transfor-mation x = u + 1 and y = v + 1 gives a correspondence between Equations 8 and 9at these parameter values. After the bifurcation points m and a vary dependentlybut we don’t know the functional relationship. The point is that each m and a increasing from the bifurcation value create a pitchfork bifurcation. HE PITCHFORK BIFURCATION 5
4. General pitchfork theorem
Recall some setting from our previous paper [4].(10) dxdt = F µ ( x ) , µ ∈ R , x ∈ R n and | µ | , | x | < ε This is associated to a family F µ with bifurcation parameter µ ∈ ( − ε, ε ) describing dxdt = F µ ( x ) . Here x belongs to a domain X of R n , and F ( x ) = F ( x ) . We supposethat the dynamics of F µ is that of a stable equilibrium x µ , basin B µ for µ < µ , and that the bifurcation is at µ . We assume that x µ = 0 is an equilibrium for all µ. Uniformity condition: ”First bifurcation from a stable equilibrium.” The equi-librium does not ”leave it’s basin” in the sense that there is a neighborhood N of x , such that N is contained in B µ for all µ < . Note this implies by a uniform continuity argument, that even at µ = 0, x is asink in the sense that x ( t ) → x , if the initial point belongs to N. The dynamicsof F µ , µ = 0 has a ”basin” B : x is a ”weak sink.” It follows that in this space B, the only equilibria of F µ is x µ for all µ ≤ . One could say that µ is the ”first”bifurcation.Define J µ to be the matrix of partial derivatives of F µ at x µ . The eigenvaluesof J µ for µ < µ all have negative real part, either real, or in complex conjugatepairs. At the bifurcation, one has either a single real eigenvalue becoming zero andthen positive with the pitchfork (if det( J µ ) = 0 then a complex conjugate pair ofdistinct eigenvalues with real parts zero becomes positive after the bifurcation andthen the Hopf oscillation occurs, not discussed here).A pitchfork bifurcation converts a stable equilibrium into two stable equilibria (theHopf bifurcation converts a stable equilibrium into a stable periodic solution). The pitchfork bifurcation theorem:
In Equation 10, let x µ be a stable equi-librium for µ for all µ < µ . Suppose the uniformity condition is satisfied. Ifthe determinant of J µ = 0 , then generically the dynamics undergoes a pitchforkbifurcation.1. If n = 1 , this has been proved in our paper [4] as discussed above.2. For n > , generically there is a pitchfork as exemplified by the normal form inSection 2. Sketched of proof of the pitchfork bifurcation theorem for n > F µ ( x ) = 0 for each µ. Consider the hypothesis above and consider the equilibrium x µ . At µ = µ , thedeterminant of J µ becomes zero. Then by the local stable manifold theory, thisequilibrium changes from a sink µ < µ , to a saddle for µ > µ . This saddle has an n − W uµ and a one dimensional expandingstable manifold W sµ . We will be using the following.
Poincar´e-Hopf index theorem : Suppose dxdt = F ( x ), x belongs to X and F : X → R n . Suppose X homeomorphic to a closed ball [8] and F ( x ) points to the INDIKA RAJAPAKSE AND STEVE SMALE interior of X for each x belonging to the boundary(11) X F ( x )=0 , x ∈ X sign (det( J )) ( x ) = ( − n where J is the Jacobian of F at x. The formula on the left side of Equation (11) isthe Poincar´e-Hopf index . In particular generically, in the case n is odd, there is anodd number of equilibria.Then we observe that the Poincar´e-Hopf index at the equilibrium for the sink is( − n for µ < µ . For µ > µ of this equilibria changes to ( − n +1 as thesink changes to a saddle. This follows from the eigenvalue structure of the saddle.Therefore, by the Poincar´e-Hopf index theorem there must be other new equilibriafor µ > µ . Generically these new equilibria must be two in number and they aresinks. We have obtained the defining property of the pitchfork.The above needs to be carried out uniformly for all µ. This procedure follows asuggestion of Mike Shub [9, 10].We consider the 2-dimensional center manifold, C M , at µ = µ and the equilibrium x = x with the added equation dµdt = 0 . This gives a dynamical system of n +1 equations. The C M is two dimensional and the stable manifold W uµ is n − J µ with negative real parts. The C M corresponds to the span of the null space (eigenvector corresponding the zeroeigenvalue) and the space of the variable µ. The C M projects on to µ and the inverseimage of µ is the expanding one dimensional manifold of the equilibrium x , for F µ .
5. A biological perspective
In our previous work [4], a cell can be thought as a point in the basin and the celltype can be identified with a basin. Thus, the identity of a specific cell type in ourgenome dynamics can be defined by characteristic gene expression pattern at theequilibrium. We suggest that the emergence of a new cell type from this originalcell type, through differentiation, reprogramming, or cancer a result of pitchforkbifurcation, is a departure from a stable equilibrium and requires cell division. Innormal cell division during differentiation or reprogramming, a cell can undergosymmetric or asymmetric division. Let A be the mother cell in the following. Insymmetric division, two identical daughter cells arise: A and A (B and B), that havegenomes with the same activity [11, 12]. In asymmetric division, two daughter cellsarise: A and B, that have genomes with different activity. Both cases above reflecta pitchfork bifurcation. In another type of asymmetric division, two daughter cellsarise, B and C, where the activity of both genomes is different from the mother alsoreflecting a pitchfork bifurcation. One example of this is in cases of abnormal celldivision, where chromosomes are mis-segregated, resulting in one daughter with toomany and one daughter with too few chromosomes. This type of division may beone of the initiating events in emerging cancer cells [13]. Capturing these events interms of our bifurcations may give us insight into emergence of a cell type.
Acknowledgments:
We would like to thank Lindsey Muir, Thomas Ried, andespecially Mike Shub for helpful discussions.
HE PITCHFORK BIFURCATION 7
References [1] Guckenheimer J, Holmes P (2002) Nonlinear Oscillations, Dynamical Systems, and Bifurca-tions of Vector Fields (Springer, New York).[2] Strogatz S (2000) Non-linear Dynamics and Chaos: With applications to Physics, Biology,Chemistry and Engineering (Perseus Books).[3] Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos (Springer-Verlag, New York, second edition).[4] Rajapakse I, Smale S (2016) Mathematics of the genome.
Foundations of ComputationalMathematics , 1-23.[5] Kuznetsov Y (1998) Elements of applied bifurcation theory (Springer-Verlag, New York,second edition).[6] Gardner T, Cantor C, Collins J (2000) Construction of a genetic toggle switch in Escherichiacoli.
Nature,
Science , 266 (5192): 1821-1828.[12] Massagu´e J (2004) G1 cell-cycle control and cancer
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18; 432 (7015): 298-306.[13] Ried T et. al. (2012) The consequences of chromosomal aneuploidy on the transcriptome ofcancer cells.
Biochim Biophys Acta.
University of Michigan
E-mail address : [email protected] City University of Hong Kong, University of California, Berkeley
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