Analysis of the Takens-Bogdanov bifurcation on m parameterized vector fields Francisco Armando Carrillo Navarro, Fernando Verduzco G., Joaquín Delgado F. Programa de Doctorado en Ciencias (Matemáticas), Departamento de Matemáticas, Universidad de Sonora.
Contenido 1 Takens-Bogdanov bifurcation 2 Statement of the problem 3 Dynamics on the center manifold 4 Main theorem 5 An example
Takens-Bogdanov bifurcation We consider a planar dynamical system ż = f (z), with f (0) = 0 and ( ) 0 1 Df (0) = J 0 0 0. (1) From normal form theory, there is a change of coordinates such that the original system can be reduced up to terms of second order to the form
ż = f 0 (z) = ( z 2 az 2 1 + bz 1z 2 ). (2) This is called the truncated normal form of the original system. Bogdanov [2] showed that the family ( ) z ż = 2 λ 1 + λ 2 z 1 + az1 2 + bz, (3) 1z 2 for ab 0 is a versal deformation of the truncated system (2).
Figure: Different dynamics of the Takens-Bogdanov bifurcation.
The point λ 1 = λ 2 = 0 separates two branches of the saddle-node bifurcation curve T +( ) = {(λ 1, λ 2 ) : λ 1 = 1 4 λ2 2, λ 2 > 0 (λ 2 < 0)}, the half-line H = {(λ 1, λ 2 ) : λ 1 = 0, λ 2 < 0} corresponds to the Hopf bifurcation that generates a stable limit cycle.
This cycle exists and remains hyperbolic between the line H and a smooth curve P = {(λ 1, λ 2 ) : λ 1 = 6 25 λ2 2+O( λ 2 3 ), λ 2 < 0}, where saddle homoclinic bifurcation occurs. When the cycle approaches the homoclinic orbit, its period tends to infinity.
Statement of the problem Consider the m-parameterized vector field ẋ = F (x, µ), (4) where x R n, µ R m, with m 2, and F C r (R n R m ), with r 2. Suppose that there exists (x 0, µ 0 ) R n R m such that H1) F (x 0, µ 0 ) = 0, and
H2) The spectrum of the linearization at the critical point satisfies σ (DF (x 0, µ 0 )) = {λ j C λ 1,2 = 0, Re(λ j ) 0, for j = 3,..., n }, the nonsemisimple case.
Our goal is to find sufficient conditions on the vector field F, such that the dynamics on the center manifold at x = x 0, is locally topologically equivalent to the versal deformation of the generic planar Takens-Bogdanov bifurcation
ż 1 = z 2 (5) ż 2 = β 1 + β 2 z 1 + az 2 1 + bz 1 z 2, where ab 0. (6)
Dynamics on the center manifold We use the center manifold theory to determine the dynamics on the m-parameterized two-dimensional center manifold at the equilibrium x = x 0 for µ µ 0.
Jordan form Let us consider the Taylor series around (x 0, µ 0 ), F (x, µ) = DF (x 0, µ 0 )(x x 0 ) + F µ (x 0, µ 0 )(µ µ 0 ) + 1 2 D2 F (x 0, µ 0 )(x x 0, x x 0 ) +F µx (x 0, µ 0 )(µ µ 0, x x 0 ) +. (7) From hypothesis H2) we have that,
The matrix A = DF (x 0, µ 0 ) R n n is similar to the matrix ( ) J0 0 J =, 0 J 1
where J 0 = ( 0 1 0 0 ), and J 1 R (n 2) (n 2) is such that σ(j 1 ) = {λ j C Re(λ j ) 0, for j = 3,..., n }
That is, there exists a invertible matrix P = (p 1 p 2 P 0 ), such that P 1 AP = J, with P 1 = q T 1 q T 2 Q 0,
Change of coordinates Let us consider the change of coordinates and parameters y = P 1 (x x 0 ), and α = µ µ 0, (8) then (7) transforms into ẏ = Jy + P 1 F µ (x 0, µ 0 )α + 1 2 P 1 D 2 F (x 0, µ 0 )(Py, Py) +P 1 F µx (x 0, µ 0 )(α, Py) +. (9)
Let us define v 0 = ( p 1 p 2 ), and w 0 = ( q ( 1 q 2 ) T, then P = ( v 0 P 0 ) and P 1 w0 =. Now we define ( y1 y 2 ) Q 0 = y = P 1 (x x 0 ) = that is, y 1 = w 0 (x x 0 ) R 2, and y 2 = Q 0 (x x 0 ) R n 2. ( w0 (x x 0 ) Q 0 (x x 0 ) ),
then (9) transforms into ẏ 1 = J 0 y 1 + w 0 F µ (x 0, µ 0 )α + 1 2 w 0D 2 F (x 0, µ 0 )(v 0 y 1, v 0 y 1 ) + w 0 D 2 F (x 0, µ 0 )(v 0 y 1, P 0 y 2 ) + 1 2 w 0D 2 F (x 0, µ 0 )(P 0 y 2, P 0 y 2 ) +w 0 F µx (x 0, µ 0 )(α, v 0 y 1 ) + w 0 F µx (x 0, µ 0 )(α, P 0 y 2 ) +
ẏ 2 = J 1 y 2 + Q 0 F µ (x 0, µ 0 )α + 1 2 Q 0D 2 F (x 0, µ 0 )(v 0 y 1, v 0 y 1 ) + Q 0 D 2 F (x 0, µ 0 )(v 0 y 1, P 0 y 2 ) + 1 2 Q 0D 2 F (x 0, µ 0 )(P 0 y 2, P 0 y 2 ) +Q 0 F µx (x 0, µ 0 )(α, v 0 y 1 ) + Q 0 F µx (x 0, µ 0 )(α, P 0 y 2 ) +.
In order to simplify this system we will use the next Definition Given ν R n, ν = L = L 1. L n product by ν 1. ν n and L R n (r s),, where L i R r s, we define the ν L = n ν i L i. i=1
Therefore, the system can be written as the extended system ẏ 1 J 0 w 0 F µ (x 0, µ 0 ) 0 y 1 α = 0 0 0 α ẏ 2 0 Q 0 F µ (x 0, µ 0 ) J 1 y 2 where F 1 (y 1, α, y 2 ) 0 F 2 (y 1, α, y 2 ) +, (10)
F 1 (y 1, α, y 2 ) = 1 [( w0 D 2 F (x 0, µ 0 ) ) (v 0, v 0 ) ] (y 1, y 1 ) 2 + [( w 0 D 2 F (x 0, µ 0 ) ) (v 0, P 0 ) ] (y 1, y 2 ) + 1 [( w0 D 2 F (x 0, µ 0 ) ) (P 0, P 0 ) ] (y 2, y 2 ) 2 + [(w 0 F µx (x 0, µ 0 )) v 0 ] (α, y 1 ) + [(w 0 F µx (x 0, µ 0 )) P 0 ] (α, y 2 ) +,
To calculate the m-parameterized local center manifold at the equilibrium point y = 0, we first consider the change coordinates where P = ξ α ζ I 2 0 0 0 I m 0 0 K 2 I n 2 = P 1 y 1 α y 2 with K 2 = J 1 1 Q 0F µ (x 0, µ 0 ). and P 1 =, (11) I 2 0 0 0 I m 0 0 K 2 I n 2,
Then, system (10) transforms into ξ J 0 R 0 0 ξ α = 0 0 0 α + ζ 0 0 J 1 ζ where R 0 = w 0 F µ (x 0, µ 0 ), and f 1 (ξ, α, ζ) = F 1 (ξ, α, ζ K 2 α), f 2 (ξ, α, ζ) = F 2 (ξ, α, ζ K 2 α). f 1 (ξ, α, ζ) 0 f 2 (ξ, α, ζ) (12)
From the center manifold theory, system (12) has a center manifold ζ = h(ξ, α) = O( ξ, α 2 ), with h(0, 0) = Dh(0, 0) = 0, and the dynamics on the center manifold is given by ξ = J 0 ξ + R 0 α + f 1 (ξ, α, h(ξ, α))
where f 1 (ξ, α, h(ξ, α)) = F 1 (ξ, α, h(ξ, α) J 1 1 Q 0F µ (x 0, µ 0 )α). Observe that it is not necessary calculate the center manifold h(ξ, α) because it does not affect the quadratic terms in f 1.
Then, we have proved the next Lemma Let the nonlinear system ẋ = F (x, µ), satisfy the non-hyperbolicity conditions H1) H2) at the equilibrium point (x 0, µ 0 ). Then the dynamics on the m-parameterized two-dimensional center manifold at the equilibrium point x = x 0 for µ µ 0, is given by
ξ = J 0 ξ + R 0 α + R 1 (α, ξ) + 1 2 R 2(ξ, ξ) where +O( α 2 ) + O( ξ, α 3 ), (13) ξ = w 0 (x x 0 ), α = µ µ 0, R 0 = w 0 F µ (x 0, µ 0 ), (14)
and R 1 = (w 0 F µx (x 0, µ 0 )) v 0 (15) + ( w 0 D 2 F (x 0, µ 0 ) ) (A 0 F µ (x 0, µ 0 ), v 0 ), R 2 = ( w 0 D 2 F (x 0, µ 0 ) ) (v 0, v 0 ), (16) with A 0 = P 0 J 1 1 Q 0.
Main theorem Then our goal is to find a change of coordinates ξ = z + L 0 α + α T L 1 z + 1 2 zt L 2 z, where L 0 R 2 m, L 1 R 2 (m 2), L 2 R 2 (2 2), such that (13) is transformed into the versal deformation of the Takens-Bogdanov bifurcation (5).
Observe that ξ = (I + α T L 1 + z T L 2 )ż, and for z 0 we have that (I +α T L 1 +z T L 2 ) 1 = I α T L 1 z T L 2 +, then, ż = J 0 z + R 0 α + α T R1 z + 1 2 zt R2 z +,
where, R 0 = J 0 L 0 + R 0, R 1 = L 1 + R 1 L 1 J 0 + L T 0 R 2 R 0 T L 2 R 2 = L 2 + R 2 2L 2 J 0, with L i = (L i2, 0) T.
Lemma There exists L 0 such that R 0 α = β 1 e 2. Proof: Observe that R 0 = w 0 F µ (x 0, µ 0 ) = ( R T 01 = ( q T 1 F µ (x 0, µ 0 ) q T 2 F µ(x 0, µ 0 ) ), R T 02 )
Then, if we define L 02 = F T µ (x 0, µ 0 )q 1, and the results it follows. β 1 = q T 2 F µ (x 0, µ 0 )α,
Lemma There exists L 2 such that 1 2 z T R2 z = (az 2 1 + bz 1 z 2 )e 2. Proof: If we define D i = q i D 2 F (x 0, µ 0 ), and ( 1 L 21 = 2 pt 2 D 2p 2 + p1 T D ) 1p 2 l12 1 l12 1 l22 1, ( p T 1 L 22 = 1 D 1 p 1 2 pt 2 D ) 2p 2 1 2 pt 2 D 2p 2 2l12 1 pt 2 D, 1p 2
where l12 1 and l 22 1 are free. The results follows with a = 1 2 pt 1 D 2 p 1, b = p1 T D 1 p 1 + p1 T D 2 p 2. (17)
Lemma There exist L 01 and L 12 such that Proof: If α T R1 z = β 2 z 1 e 2. L 01 = 1 b (( ) p T 1 D 1 p 2 + p2 T D 2 p 2 R01 + ( ) l12 1 + l22 2 R02 R11 1 R12) 2, l12 1 = 1 ( 3 pt 2 D 1 + b ) 4a D 2 p 2, and L 12 = L 11 J 0 R 11 L T 0 R 21 + R T 0 L 21,
then, R 11 1 = R 11 2 = R 12 2 = 0, and ( ) α T 0 ) R1 z = α T = (α T R1 R1 12 z 12 z 1 e 2, 1 then, if we define β 2 = α T R1 12, the results follows.
Before establish the main theorem, let us define S 1 = F T µ (x 0, µ 0 )q 2, S 2 = R 12 1 (18) [ ] 2a = b (pt 1 D 1 p 2 + p2 T D 2 p 2 ) p1 T D 2 p 2 Fµ T (x 0, µ 0 )q 1 2a 2 R1i i + R 1 b 12. (19) i=1
To unfold the double-zero bifurcation from the center manifold, the transformation T : R m R 2, given by ( ) S T T (µ) = 1 (µ µ 0 ) S2 T(µ µ 0) must have rank two, that is, S 1 and S 2 must to be linearly independent.
Theorem Given the nonlinear system ẋ = F (x, µ), (20) where x R n, µ R m with m 2, such that, there exists (x 0, µ 0 ), that satisfies the conditions
H1) F (x 0, µ 0 ) = 0, H2) σ[df (x 0, µ 0 )] = {λ 1,2 = 0; Re(λ j ) 0, j = 3,..., n }, (non-hyperbolicity) H3) ab 0, (nondegeneracy) H4) S 1 and S 2 are linearly independent, (transversality)
Then, the dynamics on the center manifold of system (20) at x = x 0 and µ µ 0, which is given by (13), is locally topologically equivalent to the versal deformation of the Takens-Bogdanov bifurcation ż 1 = z 2 ż 2 = β 1 + β 2 z 1 + az1 2 + bz 1z 2, where β 1 = S T 1 (µ µ 0), and β 2 = S T 2 (µ µ 0).
An example Consider the following system of two differential equations: ẋ 1 = x 1 x 1x 2 1+αx 1 εx 2 1, ẋ 2 = γx 2 + x 1x 2 1+αx 1 δx 2 2, (21)
The equations model the dynamics of a predator-prey ecosystem. The variables x 1 and x 2 are (scaled) population numbers of prey and predator, respectively
while α, γ, ε, and δ are nonnegative parameters describing the behavior of isolated populations and their interaction.
Assume that ε = 0 is fixed. In order to the bifurcation diagram of the system with respect to the three remaining parameters (α, γ, δ) can exhibit the codim 2 Takens-Bogdanov bifurcation in planar systems, we consider that α (0, 1) and αγ < 1.
If we consider µ = (α, γ, δ) T, then (x 0, µ 0 ) = ( ( 1 ( α, α + 1 α 2, α ) T, 1 α α(1 + α), α (1 + α) 2 ) T ) is a family of equilibrium points whose linearization has a double-zero eigenvalue,
and p 1 = ( 1 α q 1 = ( 0 1 α ) ( ) 1 + α, p 2 =, 0 ) ( ), q 2 =, 1 1+α 1 α(1+α) are the right and left (generalized) eigenvectors, respectively, associated to the eigenvalue zero.
Besides, from (17), a = 1 2 pt 1 D 2 p 1 = α 2 (1 + α) 3, b = p T 1 D 1 p 1 + p T 1 D 2 p 2 = α2 (α 1) (α + 1) 2,
and, from (19), ( ) S1 T 1 = α 4 (1 + α), 1 1 + α α 2,, α 3 ( 2(α S2 T 2 2α 1) = α 2 (α + 1) 2 (α 1), α 2 3α 2, α 2 1 α 2 ) 2α 1. α(α 1)
Thus the conditions H1), H2), H3) and H4) are satisfied for all α (0, 1). Therefore, if we choose α = 1 2, the vector field (21) is locally topologically equivalent to ż 1 = z 2, ż 2 = β 1 + β 2 z 1 + 2 27 z2 1 1 18 z 1z 2, where
with β 1 =< S 1, µ >, and β 2 =< S 2, µ > µ T = (α 1 2, γ 2 3, δ 2 9 ).
Figure: Surfaces of bifurcation: S sn surface of saddle-node bifurcation, S H surface of the Hopf bifurcation and S hom surface of homoclinic bifurcation.
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