A generalization of λ-symmetry reduction for systems of ODEs: σ-symmetries

Cicogna, Giampaolo; Gaeta, Giuseppe; Walcher, Sebastian

doi:10.1088/1751-8113/45/35/355205

Cited by 22 publications

(69 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…There is in general no algorithmic way to detect the presence of PS of a given DS.This is common feature of other ‘weak’ symmetries of DS: see for a discussion about this point in the important case of λ ‐symmetries (and of a generalization of this notion called σ ‐symmetry). We can just give a possible characterization of DS admitting a PS: assume that the DS has the following form:

{\overset{u}{true}}_{i} = f_{i} (u) = {\overset{f}{true}}_{i} (u) + h_{i} (u) p_{0} (u) (i = 1, \dots, n)

where the ‘truncated’ DS

\overset{u}{true} = \overset{f}{true} (u)

admits some vector field X = g ·∇ u as an exact symmetry and p 0 ( u ) is a scalar function invariant under X , that is

[g, \tilde{f}] = 0 and X p_{0} = 0 .

Then, one has

[g, \tilde{f}] = [g, h] p_{0} = η

in the notations of Definition 2.…”

Section: Looking For Partial Symmetries Of Dynamical Systemsmentioning

confidence: 99%

“…There is in general no algorithmic way to detect the presence of PS of a given DS.This is common feature of other 'weak' symmetries of DS: see [23] for a discussion about this point in the important case of -symmetries (and of a generalization of this notion called -symmetry ). We can just give a possible characterization of DS admitting a PS: assume that the DS has the following form:…”

Section: Looking For Partial Symmetries Of Dynamical Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Partial symmetries and dynamical systems

Cicogna

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

We start recalling the characterizing property of the 'partial symmetries' of a differential problem, that is, the property of transforming solutions into solutions only in a proper subset of the full solution set. This paper is devoted to analyze the role of partial symmetries in the special context of dynamical systems and also to compare this notion with other notions of 'weak' symmetries, namely, the -symmetries and the orbital symmetries. Particular attention is addressed to discuss the relevance of partial symmetries in dynamical systems admitting homoclinic (or heteroclinic) manifolds, which can be 'broken' by periodic perturbations, thus giving rise, according to the (suitably rewritten) Mel'nikov theorem, to the appearance of a chaotic behavior of Smale-horseshoes type. Many examples illustrate all the various aspects and situations.

show abstract

{\overset{u}{true}}_{i} = f_{i} (u) = {\overset{f}{true}}_{i} (u) + h_{i} (u) p_{0} (u) (i = 1, \dots, n)

where the ‘truncated’ DS

\overset{u}{true} = \overset{f}{true} (u)

admits some vector field X = g ·∇ u as an exact symmetry and p 0 ( u ) is a scalar function invariant under X , that is

[g, \tilde{f}] = 0 and X p_{0} = 0 .

Then, one has

[g, \tilde{f}] = [g, h] p_{0} = η

in the notations of Definition 2.…”

Section: Looking For Partial Symmetries Of Dynamical Systemsmentioning

confidence: 99%

Section: Looking For Partial Symmetries Of Dynamical Systemsmentioning

confidence: 99%

Partial symmetries and dynamical systems

Cicogna

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…The above proposition is clearly useful for constructing explicit examples of σ-symmetric DS (it is known that, given a DS, it may be very difficult to determine its σ-symmetries, because the σ-determining equations are in general differential functional equations: see [10,12] for a discussion on this aspect). for any functions µ α .…”

Section: Reduction Of Dynamical Systemsmentioning

confidence: 99%

“…All the objects (functions, vector fields) considered in this paper are assumed to be smooth enough. The presence of σ-symmetries admits interesting geometrical interpretations and algebraic aspects: for a full discussion of these arguments and several other details we refer to [10,11,12] and references therein. This is a full paper presented within ICNAAM 2012; a very short and preliminary sketch of part of these results can be found in the Enlarged Abstracts of the Conference Proceedings [13].…”

Section: Introductionmentioning

confidence: 99%

Generalized notions of symmetry of ODEs and reduction procedures

Cicogna

2013

Math. Meth. Appl. Sci.

View full text Add to dashboard Cite

This paper describes the notion of σ-symmetry, which extends the one of λ-symmetry, and its application to reduction procedures of systems of ordinary differential equations and of dynamical systems as well. We also consider orbital symmetries, which give rise to a different form of reduction of dynamical systems. Finally, we discuss how dynamical systems can be transformed into higher-order ordinary differential equations, and how these symmetry properties of the dynamical systems can be transferred into reduction properties of the corresponding ordinary differential equations. Many examples illustrate the various situations.

show abstract

“…In the case of ODEs, if we Remark 3. The symmetry relation requires the vector field to be tangent to the manifold representing the equation; this means that only integral curves of vector fields are relevant, and not the speed on these [17,19,22,63].…”

Section: Introduction Sophusmentioning

confidence: 99%

On the geometry of twisted prolongations, and dynamical systems

Gaeta¹

2020

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

I give a short review of the theory of twisted symmetries of differential equations, emphasizing geometrical aspects. Some open problems are also mentioned.Dedicated to Juergen Scheurle on the occasion of his retirement 2 G. GAETA (a = 1, ..., m); partial derivatives will be denoted by u a J , where J is a multi-index J = {j 1 , ..., j n } of order |J| = j 1 + ... + j n and(here and somewhere in the following we moved the vector index of the x for typographical convenience). We denote by u (k) the set of all partial derivatives of order k, and by u [n] the set of all partial derivatives of order k ≤ n.2.1. Geometrical description of differential equations and solutions. The x are local coordinates in a manifold B, while u are local coordinates in a manifold U ; we consider the phase manifold M = B × U , which has a natural structure of bundle (M, π 0 , B) over B with fiber U . As well known [1,13,37,57,58,70] we can associate to (M, π 0 , B) its Jet bundles J k M of any order; these have a structure of fiber bundle (J k M, π k , B) over B (with projection π k ) but also of fiber bundle over those of lower order (with projectionWe also recall that the Jet bundle is equipped with a contact structure Ω [4,33,58,68,71]; this can be encoded in the contact forms 2 ω a J := du a J − u a J,i dx i . (k) f in (J k M, π k , B), which is just the set of points (x, u [k] ) with u a = f a (x) and u a J = f a J (x). Now u = f (x) is a solution to ∆ if and only if γ(k) f ⊂ S ∆ . 4 G. GAETAstart with a complete set of DIs of order zero and one, we can generate in this way a complete set of DIs of any order, basically because if U is q-dimensional, we have k · q DIs of order k (these includes those of lower orders), as follows at once from J k M being of dimension d k = (k + 1)q + 1. The situation is different for PDEs, as the dimension of J k M grows combinatorially; see e.g. the discussion in [57].2.3. Lie-point symmetries. A (Lie-point 3 ) symmetry, or more precisely a Liepoint symmetry generator, is a vector field X on M such that its prolongation X (k) to J k M is tangent to S ∆ . For a given ∆ in the form (5), this condition is expressed in local coordinates asIn these equations, also called determining equations, the F are given and one looks for ξ, ϕ (i.e. components of the vector field X) satisfying them. As the components ψ a J of X (k) along u J are given in terms of ξ, ϕ and their derivatives, all dependencies of u a J (with |J| = 0) are fully explicit, and hence (12) decouple into a system of simpler equations, one for each monomial in the u a J ; each of these is a linear PDE for the ξ and ϕ, and they can be solved algorithmically -usually with the help of a symbolic manipulation program, as the dimension of the system can be quite large. This fails in the case of first order equations.

show abstract

A generalization of λ-symmetry reduction for systems of ODEs: σ-symmetries

Cited by 22 publications

References 27 publications

Partial symmetries and dynamical systems

Partial symmetries and dynamical systems

Generalized notions of symmetry of ODEs and reduction procedures

On the geometry of twisted prolongations, and dynamical systems

Contact Info

Product

Resources

About