Simultaneous Bifurcation of Limit Cycles and Critical Periods

The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

Nonlinear flight physics of the Lie Bracket roll mechanism

In this paper, we review the concept of Lie brackets and how it can be exploited in generating motion in unactuated directions through nonlinear interactions between two or more control inputs. Applying this technique to the airplane flight dynamics near stall, a new rolling mechanism is discovered through nonlinear interactions between the elevator and the aileron control inputs. This mechanism, referred to as the Lie Bracket Roll Augmentation (LIBRA) mechanism, possesses a significantly higher roll control authority near stall compared to the conventional roll mechanism using ailerons only; it produces more than an order-of-magnitude stronger roll motion over the first second. The main contribution of this paper is to study the nonlinear flight physics that lead to this superior performance of the LIBRA mechanism. In fact, the LIBRA performance in free flight (six DOF) is double that in a confined environment of two-DOF roll-pitch dynamics. The natural feedback from the airplane motion (roll, yaw, and sideslip) into the LIBRA mechanism boosts its performance through interesting nonlinear interplay between roll and yaw, while exploiting some of the changes in the airplane characteristics near stall.

A New Application of the $${\bar{\partial }}$$-Method

By constructing a new calculating rule of Lie bracket, we construct a new nonlinear Schrödinger hierarchy and its reduction equations via using the $${\bar{\partial }}$$ ∂ ¯ -method. Furthermore, some soliton solutions of such the equation are obtained by making use of Dirac function.

On Symmetric Brackets Induced by Linear Connections

In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with metric or symplectic structures. Given a pseudo-Riemannian metric structure, we describe the symmetric brackets induced by connections with totally skew-symmetric torsion in the language of Lie derivatives and differentials of functions. We formulate a generalization of the fundamental theorem of Riemannian geometry. In particular, we obtain an explicit formula of the Levi-Civita connection. We also present some symmetric brackets on almost Hermitian manifolds and discuss the first canonical Hermitian connection. Given a symplectic structure, we describe symplectic connections using symmetric brackets. We define a symmetric bracket of smooth functions on skew-symmetric algebroids with the metric structure and show that it has properties analogous to the Lie bracket of Hamiltonian vector fields on symplectic manifolds.

Generalized Derivations and Rota-Baxter Operators of $$\varvec{n}$$-ary Hom-Nambu Superalgebras

The aim of this paper is to generalise the construction of n-ary Hom-Lie bracket by means of an $$(n-2)$$ ( n - 2 ) -cochain of given Hom-Lie algebra to super case inducing n-Hom-Lie superalgebras. We study the notion of generalized derivations and Rota-Baxter operators of n-ary Hom-Nambu and n-Hom-Lie superalgebras and their relation with generalized derivations and Rota-Baxter operators of Hom-Lie superalgebras. We also introduce the notion of 3-Hom-pre-Lie superalgebras which is the generalization of 3-Hom-pre-Lie algebras.

Courant bracket as T-dual invariant extension of Lie bracket

We consider the symmetries of a closed bosonic string, starting with the general coordinate transformations. Their generator takes vector components ξμ as its parameter and its Poisson bracket algebra gives rise to the Lie bracket of its parameters. We are going to extend this generator in order for it to be invariant upon self T-duality, i.e. T-duality realized in the same phase space. The new generator is a function of a 2D double symmetry parameter Λ, that is a direct sum of vector components ξμ, and 1-form components λμ. The Poisson bracket algebra of a new generator produces the Courant bracket in a same way that the algebra of the general coordinate transformations produces Lie bracket. In that sense, the Courant bracket is T-dual invariant extension of the Lie bracket. When the Kalb-Ramond field is introduced to the model, the generator governing both general coordinate and local gauge symmetries is constructed. It is no longer self T-dual and its algebra gives rise to the B-twisted Courant bracket, while in its self T-dual description, the relevant bracket becomes the θ-twisted Courant bracket. Next, we consider the T-duality and the symmetry parameters that depend on both the initial coordinates xμ and T-dual coordinates yμ. The generator of these transformations is defined as an inner product in a double space and its algebra gives rise to the C-bracket.