A Study of the Rimmer Bifurcation of Symmetric Fixed Points of Reversible Diffeomorphisms

Emilia Petrişor

Abstract

We prove that the study of Rimmer bifurcation of symmetric fixed points in two–dimensional discrete reversible dynamical systems can be achieved analysing either bifurcation of critical points of a symmetric Hamiltonian function or the bifurcation of symmetric equilibrium points for a nonconservative reversible vector field. We give the normal forms for generating functions of area preserving reversible diffeomorphisms and the normal forms for nonconservative reversible vector fields associated to Rimmer bifurcation.

Mathematics Subject Classification: 58F14 Key words: reversible dynamical systems, Rimmer bifurcation

1 Introduction

In classical mechanics a lot of dynamical systems possess time–reversal symmetry, i.e. the equations of evolution are invariant under the transformation t → −t. In connection to the study of the 3–body problem, Moser [5] generalized the reversibility of a system on R^{2n}, defining time–reversal symmetry with respect to a linear reflection (involution) R (R ∘ R = id). Namely, given a dynamical system defined by the complete vector field X on R^{2n}, the system is called R– reversible if R X = −X R. Devaney [1] studied reversible dynamical systems on even dimensional compact manifolds M, reversibility being introduced by a nonlinear smooth involution R having the fixed point set, F ix(R), of dimension dim(M)/2. In the nonlinear context R–reversibility means

1.1  T R ∘ X = −X ∘ R,

where T is the tangent functor on the category of smooth manifolds. Denoting by Φ_{t} the flow of the vector field X defined on M, condition (1.1) implies:

1.2  RΦ_{t} = Φ_{−t}R, ∀t ∈ R

Balkan Journal of Geometry and Its Applications, Vol.1, No.2, 1996, pp. 87-95 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Let X be a smooth complete R–reversible vector field on R^{n}, having the flow Φ_{t}. The orbit Φ_{t}x_{0} of the point x_{0} ∈ R^{n} is called symmetric with respect to R if Φ_{t}x_{0} = R Φ_{−t}x_{0}. Taking t = 0 we get R x_{0} = x_{0}, i.e. a symmetric orbit is the orbit of a point in the set F ix(R). An equilibrium point of X is called symmetric if it lies on F ix(R). If x_{0} is a nonsymmetric equilibrium point then R(x_{0}) is also an equilibrium point.

In the discrete context, if R is a smooth involution of the smooth manifold M_{2n}, with dim(F ix(R)) = n, a diffeomorphism f of M is called R–reversible diffeomorphism if I = f ∘R is also an involution. So f = I∘R, and f^{−1} = R∘f ∘R.

Definition 1.1 Let R be a linear involution of R^{2n}. A linear map L of R^{2n} is called R–reversible if I = L ∘ R is also an involution of R^{2n}. A linear map A is called infinitesimally R–reversible if A R = −R A.

If L : R^{2n} → R^{2n} is a R–reversible linear map then λ is an eigenvalue for L iff λ^{−1} is also an eigenvalue for L.

For an infinitesimally R–reversible map A : R^{2n} → R^{2n} if λ is an eigenvalue of A of multiplicity k then −λ is also an eigenvalue of multiplicity k. Moreover, the eigenvalue 0 has even multiplicity if it occurs.

Now we are able to characterize the eigenvalues of a linear R–reversible map L : R^{2} → R^{2}: The only nontrivial linear involutions of R^{2} are S(x, y) = (−x, y), S(x, y) = (x, −y) and S(x, y) = (−x, −y). So their determinant is ±1. Since L = I ∘ R, it follows that det(L) = ±1. If L is orientation preserving then det(L) = 1, i.e. the product of their eigenvalues is 1: λλ^{−1} = 1. Therefore the normal forms for orientation preserving linear R–reversibile systems on R^{2} are:

[ \begin{matrix}λ & 0 \\ 0 & λ^{−1} \end{matrix} ] [ \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ] [ \begin{matrix}−1 & b \\ 0 & −1 \end{matrix} ] [ \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ]

If x_{0} is a fixed point of the R–reversible diffeomorphism f : M → M, where M = R^{2} or a two dimensional smooth manifold, then the linear map d_{x_{0}}f is also R–reversible. Therefore we can clasify fixed points of a two–dimensional R– reversible smooth mapping according to the properties of the eigenvalues λ, λ^{−1} of its linear part d_{x_{0}}f:

1. x_{0} is a hyperbolic fixed point if λ ∈ R, and λ ≠ 1

2. elliptic if λ, λ^{−1} ∈ C, and |λ| = 1;

3. parabolic if λ = λ^{−1} = ±1. x_{0} is called 1 : 1 resonant fixed point if the both eigenvalues are 1, and 1 : 2 resonant if the both eigenvalues are −1.

2 Bifurcation of symmetric 1 : 1 resonant fixed points in two–dimensional reversible dynamical systems

During the last decade much attention was paid to the study of the dynamics of a reversible system (see [9] and references therein) and particularly to the bifurcation of points of equilibrium of revesible vector fields [3] or bifurcation of fixed points of reversible diffeomorphisms [7].

Next we study the so called Rimmer bifurcation of a 1 : 1 resonant equilibrium/fixed point of a planar R–reversible system. Rimmer [8] studied bifurcation of a 1 : 1 fixed point in area preserving R–reversible maps using a Poincaré generating function

The Rimmer’s result states:

If (x, μ) → f(x, μ), μ ∈ I is a smooth family of area preserving R–reversible maps of an open set of the plane (R(x, y) = ±(−x, y)) and (x_{0}, μ_{0}) is a 1 : 1 resonant symmetric fixed point of f, and the Poincaré generating function satisfies some conditions, then at μ = 0 f undergoes a symmetry breaking bifurcation, that is (x_{0}, μ_{0}) is embedded in a family of symmetric fixed points that change the type from hyperbolic to elliptic or conversely, when traverse the bifurcation point, and two further families of asymmetric fixed points bifurcate from (x_{0}, μ_{0}).

The generating function involved in the treatment of Rimmer bifurcation is somewhat artifficial and the proofs are very laborious.

In the following, we shall use a different generating function suggested by Meyer [4] in the study of bifurcation of fixed points in area preserving (non-reversible) maps. In fact we establish a local correspondence between R– reversible flows and R–reversible diffeomorphism.

Let Ψ be the fractional linear transformation of C, defined by: Ψ(z) = (1 + z)(1 − z)^{−1}. Its inverse is defined as Ψ^{−1}(z) = (z − 1)(z + 1)^{−1}. It is straightforward that Ψ({z|Re(z) = 0}) = S^{1} (S^{1} is the unit circle), and the left half complex plane is mapped to the interior of the unit disc. Denote by L_{1}(R^{n}), respectively L_{−1}(R^{n}) the subset of linear transformations of R^{n} with no eigenvalue 1, respectively with no eigenvalue −1. Meyer [4] proved:

1. Ψ maps L_{1}(R^{n}) onto L_{−1}(R^{n}), and if λ_{i}, i = \overline{1, n} are eigenvalues of A ∈ L_{1}(R^{n}), then Ψ(λ_{i}) are the eigenvalues of Ψ(A).

2. If A ∈ L_{1}(R^{2n}) is a Hamiltonian matrix then Ψ(A) is symplectic, and conversely, if B ∈ L_{−1}(R^{2n}) is a symplectic transformation then Ψ^{−1}(B) is a Hamiltonian linear transformation.

Next we study the action of the map Ψ on the subspace of infinitesimally R–reversible linear maps.

Proposition 2.1 If A is an infinitesimally R–reversible linear map of R^{n} having no eigenvalue 1, then Ψ(A) is a linear R–reversible map, and conversely, if B is a linear R–reversible map having no eigenvalue −1, then Ψ^{−1}(B) is infinitesimally R–reversible. Proof. Let B = Ψ(A). Then R B R = R (I + A) (I − A)^{−1} R = R (R^{2} − R A R)(R^{2} + R A R)^{−1}= (I − A)(I + A)^{−1} = B^{−1}, that is B is R–reversible.

Conversely, let A = Ψ^{−1}(B) = (B − I)(B + I)^{−1}. B being R–reversible, it is conjugated to its inverse: B = R B^{−1} R. Hence R A = R (R B^{−1} R − R^{2})(R B^{−1} R + R^{2}) = R^{2} (B^{−1} − I) (B^{−1} + I) R = Ψ^{−1}(B^{−1}) R = −Ψ(B) R = −A R. q.e.d.

In order to discuss the bifurcation of a 1 : 1 fixed point of a R–reversible diffeomorphism we extend the action of the application Ψ to the nonlinear maps.

Let V be an open neighbourhood of 0 in R^{2} and X a R–reversible vector field defined on V having 0 as equilibrium point such that the linear vector field d_{0}X has no eigenvalue 1. Hence id − X is locally invertible in a neibourhood of 0, and we can associate to X the map f defined in that neighbourhood by f = Ψ(X) = (id + X)(id − X)^{−1}. So f is a R–reversible local diffeomorphism, having 0 as fixed point. If X depends smoothly on a parameter then so does f, and thus bifurcation of the R–reversible map f reduces to the bifurcation of the equilibrium points of X.

Exploiting this correspondence we give a simpler proof for the Rimmer bifurcations of a 1 : 1 resonant fixed point of an area preserving R–reversible diffeomorphism of the plane.

It is well known in the theory of area preserving diffeomorphisms that fixed points are critical points for a generating function. We associate a generating function in the following way: Consider a smooth area preserving R–reversible map f (R(x, y) = (x, −y)) defined on a simply connected neighbourhood of the origin of R^{2}. Suppose that (0, 0) is 1 : 1 resonant fixed point for f. Then the vector field X = (f −id)∘(f +id)^{−1} is well defined on the same neighbourhood of the origin, it is R–reversible, and has (0, 0) as double zero equilibrium point. One

verifies that the vector field Y = J^{−1} X, ( J = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )) has the property

that \frac{∂Y_{1}}{∂y} = \frac{∂Y_{2}}{∂x}. Therefore there exists a smooth real function H, such that Y = gradH. Hence the local vector field X = J Y is a hamiltonian vector field, and (0, 0) is a critical point for the Hamiltonian function H. Moreover, H ∘ R = H. Conversely, to a vector field X = JgradH, where H ∘ R = H, gradH(0) = 0, and J d^{2}H(0) has no eigenvalue 1, one associates through Ψ an area preserving R–reversible diffeomorphism f, having 0 as fixed point.

Thus the bifurcation of symmetric fixed points of f is reduced to the bifurcation of critical points of the Hamiltonian function H.

Proposition 2.2 Let f_{μ}, μ ∈ (−ϵ, ϵ) be a family of area preserving R–reversible smooth local diffeomorphisms , defined on a simply connected neighbouhood of the origin, having for μ = 0 the 1 : 1 resonant fixed point (0, 0). If the normal form of the Hamiltonian generating function of f_{μ} is H(x, y) = x^{2} − μy^{2} ± y^{4}, then the family f_{μ} undergoes a Rimmer bifurcation at μ = 0 Proof

\frac{∂H}{∂x} = 2x \frac{∂H}{∂y} = ±4y^{3} − 2μy

and

d^{2}H = [ \begin{matrix}2 & 0 \\ 0 & ±12y^{2} − 2μ \end{matrix} ]

For any μ ∈ (−ϵ, ϵ), x = 0, y = 0 is a critical point of the function H, hence an equilibrium point for the associated vector field X, and a symmetric fixed point for the local symplectomorphism f. The eigenvalues of the linear vector field A = J d_{0}^{2}H are the solution of equation:

λ^{2} + detd_{0}^{2}H = 0

For μ = 0, det(d_{0}^{2}H) = 0, hence Ψ(0) = 1 and (0, 0) is a parabolic fixed point for f.

If μ < 0, det(d_{0}^{2}H) > 0, and (0, 0) is an elliptic fixed point for f, while for μ > 0 det(d_{0}^{2}H) < 0, i.e. (0, 0) is a hyperbolic fixed point for f. Observe that when μ traverses the value 0 two assymetric fixed points (critical points for H) arise:

• in the case ” + ” and μ > 0 (0, \sqrt{μ/2}) and (0, −\sqrt{μ/2}.

The determinant of the hessian d^{2}H evaluated at these points is 8μ > 0, hence the fixed points are of elliptic type. Therefore a curve of symmetric fixed points passes through (0, 0) and further two curves of elliptic asymmetric fixed points bifurcates from (0, 0).

[figure: e; h; e; e]

• in the case ”−” and μ < 0 the two assymetric fixed points are (0, \sqrt{−μ/2}) and (0, −\sqrt{−μ/2}, and they are of hyperbolic type.

[figure: e; h; h; h]

q.e.d.

An example of a family of area preserving reversible diffeomorphisms exhibiting this type of bifurcation of symmetric periodic points is analysed in [6] in connection to disappearence/reappearence of some KAM invariant curves.

Next we show that a Rimmer type bifurcation holds for a nonconservative R–reversible diffeomorphism of the plane. In order to do that we consider a R– reversible vector field X defined on the neighbourhood of (0, 0) ∈ R^{2}, R(x, y) = (−x, y) (This choice is not a restriction because by a change of coordinates we can change the reversor).

Proposition 2.3. Let R be the reversor defined by R(x_{1}, x_{2}) = (−x_{1}, x_{2}). Then the normal forms of a R–reversible vector field X = (X_{1}, X_{2}) defined on an open set U ⊂ R^{2}, (0, 0) ∈ U, having origin as a double-zero symmetric equilibrium point are:

\begin{matrix}X_{1}(x_{1}, x_{2}) = x_{2} + O(| x |^{4}) \\ X_{2}(x_{1}, x_{2}) = x_{1} x_{2} ± x_{1}^{3} + O(| x |^{4}) \end{matrix}

Proof

R–reversibility of the vector field ensures that X_{1} is even in x_{1}, while X_{2} is odd in the same argument. Moreover the the origin being a double zero equi-

librium point the jacobian matrix is of the form [d_{0}X] = [ \begin{matrix}0 & α \\ 0 & 0 \end{matrix} ], α ≠ 0.

That is, a double zero equilibrium point of a planar reversible vector field is a codimension one equilibrium point.

The system of differential equations associated to the vector field X is:

ẋ = d_{0}X(x) + V (x),

where V = X − d_{0}X. By the change of coordinates y_{1} = x_{1}/α, y_{2} = x_{2} the

system becomes: ẏ = Jy + F (y), where J = [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] is the Jordan canonical

form of the matrix [d_{0}X]. The R–reversibility was preserved by the chosen change of coordinates. Expanding F as a Taylor series one obtains: ẏ = Jy + F_{2}(y) + F_{3}(y) + O(|y|^{4}), where the terms F_{k}(y) are terms of degree k in the Taylor expansion.

Consider the real vector space P_{k} generated by 2–vector valued monomials of degree k. Namely, if (e_{1}, e_{2}) is the standard basis in R^{2} then the corresponding basis in P_{k} is x_{k_{1}}y_{k_{2}}e_{i}, ∑_{j=1}^{2} k_{j} = k.

Our goal is to seek succesive changes of coordinates preserving R–reversibility of the system, and such that to reduce as much as possible from the k-order terms of the system. We choose the changes of the form y = z + h_{2}(z), and then z = u + h_{3}(u), where h_{k} ∈ P_{k}, k = 2, 3. In order to preserve R–reversibility, the double valued polynomials h_{k} must have the first/second component odd/even in the first argument. From the normal form theory [] it is known that can be eliminated by such changes of coordinates the terms of F_{k} that are in the image of the linear operator L_{J} : P_{k} → P_{k} defined by L_{J}(Q) = JQ − dQ JQ, where J is the above Jordan matrix. In our special case P_{k} splits as H_{e}^{o} ⊕ H_{o}^{e} (the super/subscripts e and o stand for even, respectively odd). For example the subspace H_{e}^{o} ⊂ P_{2} is generated by the two vector valued monomials:

\{E_{1} = ( \begin{matrix}z_{1}z_{2} \\ 0 \end{matrix} ) , E_{2} = ( \begin{matrix}0 \\ z_{1}^{2} \end{matrix} ) , E_{3} = ( \begin{matrix}0 \\ z_{2}^{2} \end{matrix} )\}

and the subspace H_{o}^{e} by:

\{E_{4} = ( \begin{matrix}z_{1}^{2} \\ 0 \end{matrix} ) , E_{5} = ( \begin{matrix}z_{2}^{2} \\ 0 \end{matrix} ) , E_{6} = ( \begin{matrix}0 \\ z_{1}z_{2} \end{matrix} )\}

Because we have to choose particular changes of coordinates that preserve the R–reversibility, we study only the efect of the operator L_{J} on the subspace H_{e}^{o}. It is easy to see that the image Im(L_{J}_{|H_{e}^{o}}) ⊂ H_{o}^{e}. Therefore the normal form of the considered vector field will contain only the terms of order k, k = 1, 2 that belong to the complement D_{k} of the subspace L_{J}(H_{e}^{o}) in H_{o}^{e}. By straightforward

computation we get that the terms ( \begin{matrix}0 \\ z_{1}z_{2} \end{matrix} ), ( \begin{matrix}0 \\ u_{1}^{3} \end{matrix} ) can not be reduced by

the chosen changes of coordinates.

After renaming the variables we have the normal form of the system of differential equations associated the vector field under the consideration:

\begin{matrix}ẋ = y + O(| (x, y) |^{4}) \\ ẏ = a xy + b x^{3} + O(| (x, y) |^{4}) \end{matrix}

We take the truncated normal form :

\begin{matrix}ẋ = y \\ ẏ = a xy + b x^{3} \end{matrix}

and make the rescaling :

\begin{matrix}x → α x \\ y → β y \\ t → γ t \end{matrix}

in order to get the simplest coefficients. Our system then becomes:

\begin{matrix}ẋ = \frac{γ β}{α} y \\ ẏ = γ α a xy + b \frac{γ α^{2}}{β} x^{3} \end{matrix}

Next we require

2.1  \frac{γ β}{α} = 1, i.e. γ = \frac{α}{β}

For the stability not be affected under the rescaling, α and β must have the same signs. We also require

2.2  \frac{γα^{2} b}{β} = 1,

and

2.3  γα a = 1

By (2.1), (2.3) becomes

aβ \frac{α^{2}}{β^{2}} = 1

while (2.2):

2.4  bα\frac{α^{2}}{β^{2}} = 1

If (2.4) is to hold, then a and b must have the same sign. But this is a too restrictive condition. In order to have a full generality we require bα\frac{α^{2}}{β^{2}} = ±1. So the normal form is:

\begin{matrix}ẋ = y \\ ẏ = xy ± x^{3} \end{matrix}

q.e.d.

Next we consider a candidate for a versal deformation:

\begin{matrix}ẋ = y \\ ẏ = −μ x + xy + s x^{3}, s = ±1 \end{matrix}

and study the local dynamics.

For s = 1, the origin is a symmetric equilibrium point for every μ. For μ < 0 it is of hyperbolic type, while for μ > 0 it is elliptic. When μ passes through zero (from negative to positive values) two hyperbolic asymmetric equilibrium points are born (±\sqrt{μ}, 0). In the case s = −1 when μ decreases from positive to negative values again two asymmetric equilibrium points are born: (\sqrt{−μ}, 0) – a repulsor, and (−\sqrt{−μ}, 0) an attractor. Hence a Rimmer type bifurcation occurs. Remark. One verifies that the system

\begin{matrix}ẋ = y + O(| (x, y) |^{4}) \\ ẏ = −μ x + xy + s x^{3} + O(| (x, y) |^{4}) \end{matrix}

is locally topologically equivalent near the origin to the system

\begin{matrix}ẋ = y \\ ẏ = −μ x + xy + s x^{3} \end{matrix}

Therefore () is indeed a versal deformation. Remark. Given a family f_{μ} of non-area preserving R–reversible diffeomorfisms, such that f_{0} has (0, 0) as a 1 : 1 resonant fixed point, then the associated family of R– reversible vectors fields X_{μ} = Ψ^{−1}(f_{μ}) undergoes a Rimmer type bifurcation as we have seen above. By Prop. 2.1 the family f_{μ} also undergoes the same type of bifurcation. Moreover the type of created assymetric fixed points is preserved through Ψ.

Examples of families of nonconservative diffeomorphisms exhibiting this type of bifurcation are given in [7]. Post [7] proved that Rimmer type bifurcation also occurs in non-reversible systems if only a certain order of local reversibility is satisfied.

References

[1] R.L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89–113.

[2] J. Guckenheimer, P.J. Holmes, Nonlinear oscillations, dynamical systems, and bifurcation of vector fields, Springer–Verlag, New–York, 1983.

[3] G. Ioss, M.C. Pérouème, Perturbed homoclinic solutions in reversible 1 : 1 resonance vector fields, J. Diff. Eqns., 102 (1993), 62–88.

[4] K. R. Meyer, Generic bifurcation in Hamiltonian systems, in Lect. Notes in Math. 468 (1975), 31–70.

[5] J. Moser, Stable and random motions in dynamical systems, Princeton Univ. Press, Princeton, 1973.

[6] E. Petrişor, On the dynamics of the deformed standard map, Analele Univ. din Timişoara, Vol XXXIII, Fasc.1 (1995) 91–106.

[7] T. Post, H.W. Capel, G.R.W. Quispel, J.P. Van der Weele, Bifurcation in two dimensional reversible maps, Physica A 164 (1990), 635–662.

[8] R. Rimmer, Symmetry and bifurcation of fixed points of area preserving maps, J. of Diff. Eqns. 29 (1978), 329–344.

[9] J.A.G. Roberts, G.R.W. Quispel, Chaos and time reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep. 216 (1992), 63–177.

Petrişor Emilia Department of Mathematics Technical University of Timişoara Department of Mathematics e-mail emilia@utt.ro