Extended Abstracts of the 44th Annual Iranian Mathematics Conference
27-30 August 2013, Ferdowsi University of Mashhad, Mashhad, Iran
SIMPLEST NORMAL FORM OF CONSERVATIVE
REVERSIBLE HOPF–ZERO SINGULARITY
MAJID GAZOR AND FAHIMEH MOKHTARI∗
Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan 84156-83111, Iran,
Emails: [email protected] and [email protected]
Abstract. A family of conservative reversible Hopf–Zero singularity is introduced. In this article, the simplest normal form associated with any vector field from this family with a non-degenerate
quadratic condition is presented.
1. Introduction
A not much noticed application of normal form theory is introduction of dynamically meaningful families of differential equations and
of their affiliated vector fields. An alternative for Hamiltonian systems in three dimensional state space is those with a first integral (i.e.,
conservative). The family introduced in this paper constitute “all conservative reversible systems that can be put into hypernormalization
steps”; besides, it is a “maximal Lie algebra of such systems”. Another
description is: they comprise of “all incompressible reversible systems”.
Here, we ignore the phase components and present the simplest normal forms of such systems while we assume some non-zero conditions
with regards to their quadratic parts. We skip proofs and any introduction into theory of simplest normal forms and how this family
are introduced in this article; see [1, 2, 4] for some relevant details.
2010 Mathematics Subject Classification. Primary 34C20; Secondary , 34A34,
16W5.
Key words and phrases. simplest normal form theory, conservative Hopf–Zero
singularity, reversible systems.
∗
Speaker.
1
2
GAZOR AND MOKHTARI
Another important goal of normal form theory is that normal forms
must reveal any possible symmetry and must preserve the symmetries
as the hypernormalization goes on. For instance the classical normal
form of Hopf–Zero singularity reveals a natural SO(2)-symmetry in
the three dimensional state space and any further hypernormalization
should preserves this symmetry. In this paper, the reversibility and
incompressibility are somehow considered extra symmetries that they
also must be preserved during hypernormalization.
A feedback from a IMC44 conference referee (an apparent oblivious
of our research) compelled us to highlight the following; raising a few
basic facts of the theory at the expense of the subtle nature points
related to this article. Our results here are closely related with our
original and novel results in [1, 2, 3]; taken few years of hard work. Yet
even with our already given arXiv and published results, this paper
has its own contribution (not prepared for a full journal paper). The
theory with a literature of more than a century has yet left us with
many outstanding and important open problems. The main difficulty
is due to the computational burden upon applying any smart and well
informed idea; whether works or not. Any singularity holds its own
algebraic structures and thus, its associated challenges. These make it
hard for obtaining new results. When there are extra symmetries, not
only their generic conditions must be checked and find any possible Lie
(sub)algebra structure (to guarantee the process is feasible and preserve
the symmetries), but also the transformation groups must be calculated
according to symmetry and Lie subalgebra structures. This has to be
performed before one imagines the feasibility of an extension. This
must be clear for all readers merely familiar with classical normal forms
and that these challenges require substantial efforts (even if they were
ever to work straightforwardly). These take an extensive introduction
and are skipped in here. Lastly, the importance of the introduced
family is for their vast real life applications. Hence, a mere list for
classification and normal forms (as of here) truly give rise to sufficient
standard for a descent and worthwhile contribution into the subject,
let alone a IMC44 four-pages limited conference proceeding.
A differential system
dx
+ v(x) = 0,
(1.1)
dt
where v : Rn → Rn is a smooth vector field and is called time-reversible
if there exists a reversing symmetry π : Rn → Rn such that
dπ(x)
= −v(π(x)),
dt
π −1 = π.
CONSERVATIVE REVERSIBLE HOPF–ZERO NORMAL FORM
3
For an illustrating example, dynamics of a rigid motion in spatial
space has time-reversal symmetry if and only if one may not distinguish
whether a film (taken from a solid) is played in forward or in reverse; see
[5]. Time-reversal symmetries and their correlated symmetry-breaking
(many of whom are incompressible) appear in many areas of science
such as classical mechanics, quantum mechanics, thermodynamics, and
chemical reactions. The illuminating survey paper of Lamb [5] provides
a good insight into the subject.
2. The vector field family
In this paper we are concerned with the classical normal form of
reversible Hopf–Zero singularity
∞
X
ẋ =
ai,j x2i+2 ρ2j ,
ρ̇ =
i+j=0
∞
X
bi,j x2i+1 ρ2j+1 ,
(2.1)
i+j=0
θ̇ = 1 +
∞
X
ci,j x2i+1 ρ2j+1 .
i+j=0
Here, aij , bij , cij ∈ R and π(x, ρ, θ) := (−x, ρ, θ). We will ignore the
∂
phase component. Define M := − xρ ∂ρ
from the triad generators of
a sl2 Lie algebra. Then, all such classical normal form systems (2.1)
which are conservative and incompressible are formally spanned by
(k − 2l)!
(2l+2) k −1
Rkl := l+1
ad
ρ R0 , for − 1 ≤ l ≤ k, (2.2)
4 (k + 2)! M
where
∂
R0−1 := 2ρ2 ,
∂x
n
n−1
adM v := [M, adM v] for any natural number n, and adM v := [M, v];
see [1] and the references therein for more information on sl2 -irreducible
representations of vector fields. Indeed, the vector space
Rk := span{Rkl | − 1 ≤ l ≤ k},
demonstrates an irreducible sl2 -Lie algebra representation; i.e., sl2 acts
irreducibly on Rk for each k.
The utilized notations (R-letters) here are closely related with those
of [1]; for instance R0−1 and F0−1 are basically the same but are in different coordinate systems. However, it would be inconvenient to follow
F -letters in the present setting and avoid the possible ambiguities.
4
GAZOR AND MOKHTARI
The Poisson algebra associated with first integral of the defined family are generated by
X
r(x, ρ) :=
alk xl+1 ρ2(k−l+1)
(2.3)
and are equipped with the usual Poisson bracket, where −1 ≤ l ≤
k and l + k ≥ 1. Thereby, the conservative system considered in this
paper are affiliated with the vector field
X
−1
R(1) = a−1
R
+
alk Rkl .
(2.4)
0
0
This establishes the well-defined vector fields and their affiliated Lie
algebra structure that are needed for deriving normal forms.
3. Normal form theorem
In this section we present the simplest normal form of differential
systems associated with R(1) .
Theorem 3.1. Let R(1) follow Equation (2.4) and a−1
6= 0. Then,
0
there exist invertible, incompressible and reversible-preserving transformations such that they transform R(1) into its infinite level normal
form
∞
X
R(∞) = a0 R0−1 +
ak Rkk , a0 = a−1
(3.1)
0 .
k=1
The equivalent system of v
(∞)
in the cylinder coordinates is given by
2
ẋ = a0 ρ + a1 x2 + a2 x4 + a3 x6 + · · · ,
ρ̇ = −a1 xρ−2a2 x3 ρ − 3a3 x5 ρ + · · · .
(3.2)
(3.3)
(Recall that the phase component is ignored). Further, the infinite level
normal form of the first integral associated with (3.2)–(3.3) is
2 a0 2
2
4
6
ρ + a1 x + a2 x + a3 x + · · · .
r(x, ρ) := ρ
2
References
1. M. Gazor and F. Mokhtari (2013) Normal forms of Hopf–Zero singularity, arXiv
preprint arXiv:1210.4467v4.
2. M. Gazor and F. Mokhtari (2013) Volume-preserving normal forms of Hopf–Zero
singularity, arXiv preprint arXiv:1203.1807v4.
3. M. Gazor, F. Mokhtari, and J.A. Sanders (2013) Normal forms for Hopf–Zero
singularities with nonconservative nonlinear part, J. Differential Equations 254
1571–1581.
4. M. Gazor and P. Yu (2012) Spectral sequences and parametric normal forms, J.
Differential Equations 252 1003–1031.
5. J.S.W. Lamb and J.A.G. Roberts (1998) Time-reversal symmetry in dynamical
systems: a survey, Phys. D 112 1–39.