Analysis of the TakensBogdanov bifurcation on m parameterized vector fields


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1 Analysis of the TakensBogdanov 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.
2 Contenido 1 TakensBogdanov bifurcation 2 Statement of the problem 3 Dynamics on the center manifold 4 Main theorem 5 An example
3 TakensBogdanov bifurcation We consider a planar dynamical system ż = f (z), with f (0) = 0 and ( ) 0 1 Df (0) = J (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
4 ż = f 0 (z) = ( z 2 az 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).
5 Figure: Different dynamics of the TakensBogdanov bifurcation.
6 The point λ 1 = λ 2 = 0 separates two branches of the saddlenode bifurcation curve T +( ) = {(λ 1, λ 2 ) : λ 1 = 1 4 λ2 2, λ 2 > 0 (λ 2 < 0)}, the halfline H = {(λ 1, λ 2 ) : λ 1 = 0, λ 2 < 0} corresponds to the Hopf bifurcation that generates a stable limit cycle.
7 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.
8 Statement of the problem Consider the mparameterized 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
9 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.
10 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 TakensBogdanov bifurcation
11 ż 1 = z 2 (5) ż 2 = β 1 + β 2 z 1 + az bz 1 z 2, where ab 0. (6)
12 Dynamics on the center manifold We use the center manifold theory to determine the dynamics on the mparameterized twodimensional center manifold at the equilibrium x = x 0 for µ µ 0.
13 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 ) 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,
14 The matrix A = DF (x 0, µ 0 ) R n n is similar to the matrix ( ) J0 0 J =, 0 J 1
15 where J 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 }
16 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,
17 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 P 1 D 2 F (x 0, µ 0 )(Py, Py) +P 1 F µx (x 0, µ 0 )(α, Py) +. (9)
18 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 ) ),
19 then (9) transforms into ẏ 1 = J 0 y 1 + w 0 F µ (x 0, µ 0 )α 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 ) 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 ) +
20 ẏ 2 = J 1 y 2 + Q 0 F µ (x 0, µ 0 )α 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 ) 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 ) +.
21 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
22 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 = 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)
23 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 ) +,
24 To calculate the mparameterized local center manifold at the equilibrium point y = 0, we first consider the change coordinates where P = ξ α ζ I 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 I m 0 0 K 2 I n 2,
25 Then, system (10) transforms into ξ J 0 R 0 0 ξ α = 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)
26 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))
27 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.
28 Then, we have proved the next Lemma Let the nonlinear system ẋ = F (x, µ), satisfy the nonhyperbolicity conditions H1) H2) at the equilibrium point (x 0, µ 0 ). Then the dynamics on the mparameterized twodimensional center manifold at the equilibrium point x = x 0 for µ µ 0, is given by
29 ξ = J 0 ξ + R 0 α + R 1 (α, ξ) 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)
30 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.
31 Main theorem Then our goal is to find a change of coordinates ξ = z + L 0 α + α T L 1 z 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 TakensBogdanov bifurcation (5).
32 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 zt R2 z +,
33 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.
34 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 )
35 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,
36 Lemma There exists L 2 such that 1 2 z T R2 z = (az 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 pt 2 D 2p 2 2l12 1 pt 2 D, 1p 2
37 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)
38 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 + ( ) l 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,
39 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.
40 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
41 To unfold the doublezero 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.
42 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
43 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 }, (nonhyperbolicity) H3) ab 0, (nondegeneracy) H4) S 1 and S 2 are linearly independent, (transversality)
44 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 TakensBogdanov 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).
45 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)
46 The equations model the dynamics of a predatorprey ecosystem. The variables x 1 and x 2 are (scaled) population numbers of prey and predator, respectively
47 while α, γ, ε, and δ are nonnegative parameters describing the behavior of isolated populations and their interaction.
48 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 TakensBogdanov bifurcation in planar systems, we consider that α (0, 1) and αγ < 1.
49 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 doublezero eigenvalue,
50 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.
51 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,
52 and, from (19), ( ) S1 T 1 = α 4 (1 + α), 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)
53 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 z z 1z 2, where
54 with β 1 =< S 1, µ >, and β 2 =< S 2, µ > µ T = (α 1 2, γ 2 3, δ 2 9 ).
55 Figure: Surfaces of bifurcation: S sn surface of saddlenode bifurcation, S H surface of the Hopf bifurcation and S hom surface of homoclinic bifurcation.
56
57 Referencias Bazykin, A. Mathematical Biophysics of Interacting Populations. Nauka, Mmoskow. In Russian Bogdanov, R.I. Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues. Funct. Anal. Appl. 9, Carrillo, F.A. and Verduzco, F. Control of the Planar TakensBogdanov Bifurcation with Applications. Acta Appl. Math. 105: Guckenheimer, J., Holmes, P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, Vol. 42. SpringerVerlag Kuznetsov, Y.A. Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, Vol Springer. Second Edition Takens, F. Forced osccilations and bifurcations. Comm. Math. Inst. Rijkuniversiteit Utrecht, 3, S. Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Applied Mathematics, Vol. 2. SpringerVerlag. Second Edition
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