8.1 Bifurcations of Equilibria


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1 1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integrodifferential equations discrete mappings etc Of course we are concerned with ODEs Local bifurcations refer to qualitative changes occurring in a neighborhood of an equilibrium point of a differential equation or a fixed point of an associated Poincaré map These can be studied by expanding the equations of motion in power series about the point Consider an ODE depending on a parameter : Assume this system has an equilibrium point that is a sink for where is a critical value of the parameter Graph the evls as a function of Real and imaginary axes region of evls with negative real parts for arrows showing that the real parts of some evls are increasing with A real evl or a cc pair may traverse the imaginary axis as increases through The sink becomes a source or a saddle Or the equilibrium point may simply disappear when it has a zero evl! Example Fishery model with constant harvesting Recall the logistic model of population growth with an additional constant term Interpret as a model of a fishery with the number of fish time the fish growth rate the carrying capacity and a constant rate of harvesting The model may be simplified by measuring the quantity of fish and the harvest rate in units of the carrying capacity: and Further simplify by introducing a dimensionless time variable: Then obtain [1] Equilibria correspond to Analyze graphically
2 2 sketch   The direction of the arrows on the axis now depends on the sign of When there are two equilibrium points when there is one and when there are none Alternately we can study the equilibria algebraically Let Then equilibrium points of [1] correspond to or The equilibria are These exist only for add arrows to graph We can write [1] in the form [2] where the dot now refers to the derivative with respect to dimensionless time The eigenvalues of the equilibrium points are which depends on through Since its eigenvalue is negative and is a stable equilibrium point Since its eigenvalue is positive and is an unstable equilibrium point As increases toward the equilibria converge and their eigenvalues approach 0 At the equilibria merge as a single nonhyperbolic equilibrium point and for they disappear Draw a bifurcation diagram for the logistic model with constant harvesting by plotting the equilibria as a function of sketch   the two branches of the saddlenode A saddlenode bifurcation occurs at a critical value of the parameter Simplify the logistic model with constant harvesting by centering the bifurcation at the origin Let and Then [2] becomes [3]
3 3 This is the normal form of the saddlenode bifurcation For there are two hyperbolic equilibrium points for there is a single nonhyperbolic equilibrium and for there are no equilibria The saddlenode bifurcation requires 3 conditions on the vector field: Singularity condition (the equilibrium point [3] is nonhyperbolic at ) Nondegeneracy condition (the coefficient of in [3] is nonzero) Transversality condition (that guarantees that the parameter perturbs the nonhyperbolic equilibrium point in a transverse way: in [3]) Local Bifurcations of Limit Cycles To further illustrate the meaning of local bifurcation let s briefly describe local bifurcations of limit cycles sketch limit cycle section intersection Consider a Poincaré map corresponds to a fixed point of : The eigenvalues of are the Floquet multipliers (not including the trivial unit multiplier) Real and imaginary axes region of evls within the unit circle for arrows showing that evls can traverse the circle though or a complex conjugate pair can pass out of the unit circle off the real axis Example A saddlenode bifurcation of limit cycles Suppose that for there is a semistable limit cycle and an associated Poincaré map: sketch semistable limit cycle in 2D and nearby trajectories sketch   the line the Poincaré map is concave down and tangent to the line at the origin cobweb paths inside and outside the limit cycle sketch pair of limit cycles The outer one is stable and the inner one unstable Nearby trajectories sketch   the line the Poincaré map has shifted so there are now two intersections with the line Three cobweb paths
4 4 sketch spiral flow without limit cycle no intersection between the Poincare map with the line The saddle node bifurcation corresponds to a single eigenvalue reaching the unit circle at and then disappearing Global Bifurcations of Limit Cycles Global bifurcations cannot be described by a local analysis Example Homoclinic bifurcation in sketch like Meiss Fig 818 and without the point and axis Notice the limit cycle for 82 Preservation of Equilibria Consider An equilibrium point satisfies When we linearize about the equilibrium point the Jacobian matrix is If the Jacobian matrix has a zero eigenvalue then it is singular and we say the equilibrium point is degenerate Otherwise the Jacobian matrix is nonsingular and the equilibrium point is nondegenerate Contrast this with the definition of a nonhyperbolic equilibrium point where only the real part of an eigenvalue need be zero For example if a Jacobian matrix has a pair of complex conjugate imaginary eigenvalues and all of the other eigenvalues are nonzero it is not singular Example Fishery model with constant harvesting One way that we have written the equation of motion is Let be an equilibrium point When and the equilibrium point is degenerate; corresponds to the saddlenode bifurcation For the equilibrium point disappears
5 5 If an equilibrium point is nondegenerate it cannot be removed by sufficiently small perturbations (such as a small change in the value of Theorem 81 (Implicit Function) Let be an open set in and with Suppose there is a point such that and is a nonsingular matrix Then there are open sets and and a unique function for which and The proof of this famous theorem probably appears in your favorite analysis book To gain a rough understanding of why the condition on the Jacobian is necessary expand about : Neglect the higherorder terms and solve for to obtain This can be done for arbitrary only if is nonsingular When discussing the system the application of the implicit function theorem to preservation of equilibria corresponds to and Corollary 82 (Preservation of a Nondegenerate Equilibrium) Suppose the vector field is in both and and that is a nondegenerate equilibrium point for parameter value Then there exists a unique curve of equilibria passing through at Proof The matrix governs the stability of the equilibrium and since has all of its eigenvalues nonzero is nonsingular Then theorem 81 implies that there is a neighborhood of for which there is a curve of equilibria 83 Unfolding Vector Fields Change of Variables and Topological Conjugacy Recall topological conjugacy between flows corresponds to the diagram:
6 6 where is a homeomorphism The diagram implies Bifurcation theory studies systems that depend on parameters Let be the flow of and let be the flow of If these flows are conjugate for each value of then there is the diagram: where is a homeomorphism The diagram implies Example Consider the system Change variables using to obtain Note that is a homeomorphism (in fact a diffeomorphism) The corresponding flows are topologically conjugate Meiss extends this terminology to the vector fields; he also calls and conjugate For a fixed value of gives a vector field For all possible values of gives a family of vector fields A family of vector fields is induced by a family if there is a continuous map such that If the family is induced by the family then has the same dynamics as or is simpler than Example The family is induced by the family using the map The family has the same dynamics as (In fact for corresponding values of and the vector fields are the same) Example Consider the family and the constant map The family of vector fields induced by using the map is The family is clearly simpler than the family Let be the flow of and be the flow of where the vector fields and are conjugate for corresponding values of and Then we have the diagram
7 7 where and Then the flows are related by [1] If is a diffeomorphism we can also find a relationship between the corresponding vector fields Differentiate [1] with respect to : Set to obtain the desired relationship [2] where and Compare with Eq 434 in Meiss Example Let and where and Then and [2] gives This can be verified by making the indicated substitutions for Unfolding Vector Fields and Consider a vector field that has a degenerate orbit This could be a degenerate equilibrium point or for example the homoclinic orbit in the homoclinic bifurcation Then we say fulfills a singularity condition A family of vector fields is an unfolding of if Example The singularity associated with the saddlenode bifurcation is is degenerate because it has a zero eigenvalue Two unfolding of are and
8 8 An unfolding of is versal if it contains all possible qualitative dynamics that can occur near to This means that every other unfolding in some neighborhood of will have the same dynamics as some family induced by Example Let and We will show below that is a versal unfolding of the saddle node bifurcation However is not versal For example there is no value of for which the system has two equilibrium points An unfolding is miniversal if it is a versal unfolding with the minimum number of parameters Example Let and The flows of and are diffeomorphic To see this complete the square in : The change of variables and converts into Note that is a diffeomorphism Since may be any real number the families of vector fields and have the same dynamics However has two parameters and has only one Then is a versal unfolding but not a miniversal unfolding of the saddle node bifurcation 84 SaddleNode Bifurcation in One Dimension Consider the ODE on Theorem 83 Suppose that with a nonhyperbolic equilibrium at the origin and that satisfies the nondegeneracy condition Then there is a such that when there is an open interval containing such that there is a unique extremal value There are two equilibria in when one when and zero when Proof Let The singularity and nondegeneracy conditions imply
9 9 Since the higher order term satisfies A general unfolding of will have form [3] where and Consider the function Since the implicit function theorem guarantees neighborhoods and of the origin such that when there is a unique such that and Since there must also be an open interval containing for which is monotone On is either concave up (when ) or concave down (when ) Therefore is the unique extremal value for when and determines whether the critical point is a minimum or a maximum Since when this remains true by continuity for small enough ; for Therefore when has a minimum at and is positive on This implies that if there are no zeros of if there is a single zero and if there are two zeros Similar considerations apply when Picture for :   For is a stable equilibrium and is an unstable equilibrium The bifurcation set is the set of all points in parameter space at which the bifurcation takes place A bifurcation is codimension if the bifurcation set is determined by independent conditions on parameters According to theorem 83 the saddlenode bifurcation depends on a single quantity: The bifurcation takes place when The saddlenode is a codimension 1 bifurcation Example Let Find the saddlenode bifurcation set that is the locus of saddlenode bifurcations in space is defined by From we have Then The bifurcation set corresponds to the single condition
10 Corollary 84 If f satisfies the hypotheses of theorem 83 and there is a single parameter such that the transversality condition holds then a saddlenode bifurcation occurs as crosses zero Proof Recall that was defined to satisfy We also defined Thus the chain rule implies 10 Since the sign of changes as crosses zero Remark When there is a vector of parameters application of the corollary requires that for as crosses Example Let If a saddlenode bifurcation takes place as crosses Otherwise the bifurcation takes place when Example Let an unfolding of Then and By the corollary a bifurcation occurs as crosses zero For this system and a saddlenode bifurcation also takes place when Example Let an unfolding of Note that which does not satisfy the transversality condition In fact is an equilibrium of for all values of A saddlenode bifurcation clearly does not occur in this unfolding of Theorem 85 Under the hypotheses of theorem 83 the saddlenode bifurcation has the miniversal unfolding [1] Proof Expand in Taylor series about : Identifying and gives [2] Let and Then [3]
11 11 Compare [2] and [3] with the form of section 83 Eq [2] which is reproduced below To obtain that equation we assumed that the flows of and were related by the diffeomorphism where Then we obtained the relationship between vector fields Note that the expression for given below [2] is a diffeomorphism and that Eq [3] exhibits the correct constant of proportionality [3] gives The comparison between [2] and Thus the family of vector fields is induced by a family of form [4] According to the onedimensional equivalence theorem theorem 410 there is a neighborhood of the origin for which the dynamics of [4] is topologically equivalent to those of [1] because both systems have two equilibria with the same stability types and arranged in the same order on the line Transcritical Bifurcation Consider an unfolding of This is not a versal unfolding; the equilibrium point at never disappears is a normal form of the transcritical bifurcation Sketch the bifurcation diagram   two lines of equilibria with stability exchanged at the origin   with parabolas intersecting the abscissa at and to determine stability of equilibrium points This bifurcation is sometimes referred to as an exchange of stability between the two equilibrium points and is encountered frequently in applications To see the relationship with the saddlenode bifurcation begin by completing squares: Let and to see that that flow of is homeomorphic to that of The family of vector fields is induced by using
12 12 Notice that cannot be positive As increases through zero values of take the following path through the saddlenode bifurcation diagram:   curves of equilibria corresponding to SN bifurcation path of As a bug crawls along this path it sees the transcritical bifurcation diagram shown above (modulo some distortion due to the coordinate transformations) 86 SaddleNode Bifurcation in Theorem 86 (saddle node) Let and suppose that satisfies (singularity) Choose coordinates so that is diagonal in the zero eigenvalue and set where corresponds to the zero eigenvalue and are the remaining coordinates Then where and Suppose that (nondegeneracy) Then there exists an interval containing functions and and a neighborhood of such that if there are no equilibria and if there are two Suppose that has a dimensional unstable space and an dimensional stable space Then when there are two equilibria one has a dimensional unstable and an dimensional stable manifold and the other has a dimensional unstable manifold and an dimensional stable manifold Proof The equilibria are solutions of [1] By assumption that there is a neighborhood of such that is nonsingular; thus the Implicit Function Theorem ensures where there exists a unique function
13 13 [2] and Substitute this into to obtain Consequently the problem has been reduced to the onedimensional case; we need only check that satisfies the same criteria as Theorem 83 the onedimensional case It is easy to see that Since is so is and differentiation of [2] with respect to gives Since this implies that This relationship helps compute the required derivatives of : Thus the needed hypotheses for Theorem 83 are satisfied and there exists an extremal value such that when crosses zero the number of equilibria changes from zero to two The stability of the equilibria follows by considering the stability of equilibria in the one dimensional case in conjunction with the nonhyperbolic HartmanGrobman theorem from our earlier studies of the center manifold Now we can see why this is called a saddlenode bifurcation Consider theorem 86 for sketch in 1D case; two equilibria for degenerate equilibrium for no equilibria for sketch for : for a stable node and saddle; for a degenerate node; for a converging flow without equilibrium sketch for : for a saddle and an unstable node; for a degenerate node; for a diverging flow without equilibrium Tranversality The following theorem gives a condition that guarantees that parameter is varied changes sign as a
14 14 Corollary 87 (transversality) Assume that satisfies the hypotheses of theorem 86 If is any single parameter such that (transversality) then a saddlenode bifurcation takes place when crosses zero Proof Show that Use to denote the critical point of as a function of Then and The first derivatives and both vanish by hypothesis then the transversality assumption gives Example Let s apply the systematic procedure suggested in the statement of theorem 86 to the system which has an equilibrium at the origin First rewrite this in matrix form [3] The Jacobian matrix has eigenvalues and Corresponding eigenvectors are and The matrix equation has the general form where Let the transformation matrix Then Set and We have or
15 15 This has the form given in the statement of theorem 86 (see [1]) The system satisfies the singularity conditions stated in the theorem and the nondegeneracy condition Furthermore so the transversality condition is satisfied Since has a minimum and since the minimum decreases through as increases Going back to the original system [3] we can easily solve for the equilibria to get confirming our result and 85 Normal Forms In chapter 2 we transformed linear systems in order to put them in a simple form (diagonalizing the matrices in the semisimple case) In chapter 4 we learned that these changes of coordinates were diffeomorphisms In this section we continue the program of applying diffeomorphisms to transform to simpler forms only now we seek to simplify the nonlinear terms Homological Operator Let and have an equilibrium point at the origin Further assume has as many derivatives as necessary for the manipulations below Expand in power series [1] where is a vector of homogeneous polynomials of degree Let A basis for is the set of monomials
16 16 where and This compact notation is known as multiindex notation For example is three dimensional Let ( factors) be the space of vectors of homogeneous polynomials on Example has dimension 6 and the basis Denoting the standard basis vectors of by the vector monomials provide a basis for Example let and The first degree terms in the power series [1] may be written We will construct the simplest vector field that is diffeomorphic with by a near identity transformation Let represent the new variable so that and the diffeomorphism is In a small neighborhood of the equilibrium point of at the origin is close to the identity transformation and is invertible Recall from chapter 4 that if generates the flow and if generates the flow then
17 17 If there is a diffeomorphism then between the two flows so that or Set to obtain a relationship between the vector fields [2] We will choose to eliminate as many of the nonlinear terms in as possible First consider the quadratic terms setting Write We will attempt to choose so as to eliminate so that Substituting the expansions for and into [2] we find or The linear terms on the two sides of the equation match Equating the quadratic terms gives [3] which is an equation for the unknown function is called the homological operator is a linear operator on the space of degree vector fields: (see problem 6) Eq [3] can be solved if and only if where is the range of Consequently we introduce the following direct sum decomposition of : where is a complement to and we split into two parts:
18 18 The function is the resonant part of We now reconsider the derivation of Eq [3] only aiming to eliminate the nonresonant terms Let Then insert the new expansions for and into [2] to obtain which is guaranteed to have a solution for Matrix Representation Prior to applying the near identity transformations we assume that coordinates have been transformed to bring the matrix into Jordan form Let be the standard unit basis vectors in this coordinate system Any linear operator on a finite dimensional space has a matrix representation Suppose where is a vector space of homogeneous polynomials and and let of basis vectors: represent a basis for Then and is given by a linear combination This defines the matrix as a representation of the action of on Writing and we have which implies Since the basis vectors are linearly independent this is equivalent to the matrix equation The simplest case is when the eigenvalues of are real and distinct Then in Jordan form Compute the action of on the monomial basis vectors of From [3]
19 19 Since is diagonal and is proportional to we have Only the th component of is nonzero and therefore only the th row of the Jacobian matrix is nonzero: Then is the dot product of this th row with the vector The th term is this dot product is Then Thus we have The vector monomials are eigenfunctions of on with eigenvalues If all the are nonzero we can solve by inverting to obtain Example Consider the 1D case with a hyperbolic equilibrium: : Set The homological operator is so with Since the nonlinear terms are all proportional to with there are no resonant terms and all nonlinear terms may be eliminated to obtain [4] This results is consistent with the
20 20 HartmanGrobman theorem which tells us that the dynamics of [4] in a neighborhood of the origin are topologically conjugate to those of the linearized system When one or more of the are zero is nontrivial and there may be resonances For example consider the case of for which the eigenvectors of were found to be Suppose Then the condition corresponds to For the case of and Then If then is nontrivial and may have resonant nonlinear terms that cannot be eliminated This may not be a surprise since is not hyperbolic For the case of and Then Notice that this can be zero even if is hyperbolic In other words certain nonlinear systems with hyperbolic linear parts may have resonant nonlinear terms that cannot be eliminated by normal form transformations Compare this to the statement of the HartmanGrobman theorem which assures us that the dynamics of a nonlinear system in a neighborhood of a hyperbolic equilibrium point are topologically conjugate to those of the corresponding linearized system This difference reflects the fact that normal form transformations use diffeomorphisms to eliminate the nonlinear terms This is a smaller class of transformations than the homeomorphisms used by the HartmanGrobman theorem The nonhyperbolic HartmanGrobman theorem from center manifold theory tells us that the dynamics of a nonlinear system are topologically conjugate to the linearized dynamics on the stable and unstable manifold together with the nonlinear dynamics on the center manifold Thus we are particularly interested to apply normal form transformations to systems with nonhyperbolic linear parts Example Doublezero eigenvalue Consider a 2D system of form [5] The Jacobian for this system evaluated at the origin is
21 21 is the most typical Jordan form for a system with two zero eigenvalues The alternative is identically zero We attempt to eliminate the quadratic terms on the right hand side of [5] using a near identity transformation of form where and Denote The homological operator is Recalling etc we have etc In this way we construct a matrix representation for in the basis The result is The column space of defines its range The resonant space is any subspace complementary to Since it must be an element of The second vector may be any linear combination of and that is independent of The simplest choices for are and Using the second choice leads to [6]
22 22 This second form has the advantage that it is equivalent to the nonlinear oscillator The right hand side of [6] is the singular vector field whose unfolding is known as the Takens Bogdanov bifurcation which Meiss discusses in section 810 Higher Order Normal Forms Proceed by induction Suppose that all terms in the range of order To this order have been eliminated below where contains the resonant terms through order Now let and require that the vector field for have only resonant terms through order Then Substitute these expansions into the relationship between diffeomorphic vector fields that we reproduce to obtain where the homological operator is on the left hand side of the equation Set [7] and equate terms in [7] of order to obtain This equation may be solved since the right hand side is in the range of
23 Example Continue the normal form transformations for the double zero (TakensBogdanov) bifurcation to third order to obtain See Meiss Chapter 8 problem 18 23
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