Hypernormal form theory: foundations and algorithms

Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

Honeycomb sandwich plate have been widly applied in industry design in recent years. In this paper, we study the cubic hypernormal form (the simplest normal form and the unique normal form) for honeycomb sandwich plate dynamics model with the help of Maple symbolic computation. Firstly, we get the average equation of four dimensional cartesian form by using the method of multiple scales perturbation analysis. Based on the method combined new grading function with multiple Lie brackets, we obtain the hypernormal form of cubic truncated. The results will further enrich the research for dynamics of honeycomb sandwich plate model, and is also the basis for higher order normal form research.

Among all bifurcation behaviors of analytic parametric families of real planar vector fields, those that stand out most prominently are confluences of distinct stationary points. The qualitative change is so drastic that in some classes of families (e.g., fold-like bifurcations) the stationary points leave the real plane altogether and slip into the complex plane. Although they disappear from the real domain they continue to organize the dynamics, and studying complex planar vector fields becomes a necessity even for real bifurcations.Our main concern is to describe a la Martinet-Ramis the analytical classification of generic holomorphic families unfolding a saddle-node vector field, and to relate this classification both to the dynamics of individual members of the family and to analytic properties of the saddle-node. For instance the problem of the existence of an analytic center-manifold for the saddle-node is characterized in terms of persistence (as the parameter tends to the bifurcation value) of heteroclinic connections between stationary points.We emphasize the geometric aspect of the classification. Complex trajectories are connected real surfaces allowing for richer geometric constructions as compared to 1-dimensional real trajectories. The trajectories are split by a finite collection of open “fibred squid sectors,” attached by spirals to stationary points within their adherence. The sectors are carved in such a way that one can construct an analytic and bounded conjugacy between the vector field and its formal normal form. The invariants of classification are obtained as transition maps of overlapping such normalization charts. Since we can perform this sectorial normalization analytically in the parameter, by restricting its values to “cells” covering the parameter space minus the bifurcation value, the resulting finite collection of functional invariants is analytic on parameter cells and continuous on their adherence. In that sense it “unfolds” Martinet-Ramis invariant of the saddle-node.The inverse problem (or realization) is addressed in the case of a persistent heteroclinic connections and provides unique normal forms (universal family for the analytic classification). We particularly show that in general the invariant cannot depend holomorphically on the parameter over a full neighborhood of the bifurcation value.

Unique normal forms for lambda calculus with surjective pairing

Application of nonlinear dynamic analysis in the identification and control of nonlinear systems—II. More complex dynamics

Based on wings flutter on flying aircraft in this paper, the authors study the mechanical model of the rectangular symmetric cross-ply composite laminated plates. Frist, the method of multiple scales is employed to obtain the four-dimensional averaged equations of the model. Then, the method of new grading function and multiple Lie brackets is utilized to obtain the hypernormal form (simplest normal form, unique normal form) at cubic of above averaged equations.

We have studied a variety of weakly perturbed nonlinear dynamical systems using the method of normal forms, a reduction scheme, introduced by Poincare in the late nineteenth century. The method was formalized by Birkhoff who applied it extensively to Hamiltonian mechanics. By invoking a near-identity coordinate transformation, the method of normal forms converts the nonlinear differential equations into simplified equations of motion for the zerothorder approximation to the true solution [1,2,3]. These coordinate transformations, nonlinear functions of the zeroth-order approximations, are found by solving a sequence of linear equations which are determined by the spectrum of the operator associated with the linear, unperturbed motion. The algebra related to these calculations is intensive and well-suited to the symbolic and programming capabilities of Maple. Through the use of procedures written in Maple we have performed high-order normal form computations and have investigated the exploitation of an inherent nonuniqueness of the zeroth-order approximation [4,5]. We will show how this nonuniqueness can be utilized to obtain transformed equations of motion that can be tailored to the needs of the investigator. Coupled with computer algebra, the method of normal forms is a potentially powerful tool for the examination of nonlinear systems [6].KeywordsNormal FormComputer AlgebraDuffing EquationUnperturbed MotionAnalyze Nonlinear SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper investigates observer design problem for a large class of nonlinear singular systems with multi outputs. We firstly regularize the singular system by injecting the derivative of outputs into the system. Then differential geometric method is applied to transform the regularized system into a simple normal form, for which a simple observer is proposed.

Integrability and linearizability of cubic Z2 systems with non-resonant singular points

We present a deduction method for propositional modal logics. It is based on a resolution principle for formulas written in a very simple normal form, close to the clausal form for classical logic. It allows us to extend naturally resolution and refinements of resolution to modal logics.

In computer algebra, it remains to be challenging to establish general computational theories for determining the equivalence of indexed polynomials. In previous work, the author solved the equivalence determination problem for Riemann tensor polynomials by extending Grobner basis theory. This paper extends the previous work to more general indexed polynomials that involve no eliminations of indices and functions, by the method of ST-restricted rings. A decomposed form of the Grobner basis of the defining syzygy set in each ST-restricted ring is provided, and then the canonical form of an indexed polynomial proves to be the normal form with respect to the Grobner basis in the ST-fundamental restricted ring.

Calculating normal-forms is one of the main algorithms on studying nonlinear dynamical system. There are matrix-expressing, conjugate-operator and Lie-algebra algorithms which can be used in the calculation. It is well known that calculating with conjugate-operator is the main step in finding all orders of normal forms. In this paper, a Mathematica package is presented for calculating normal- forms with inner-product algorithm. The package we compiled possesses generality in calculating high order normal-forms of nonlinear dynamical systems.

The truncation theorem for the reduction of unfoldings to normal form to specified degree in the control variables is generalized to unfoldings of singularities having non-zero rank, by using the splitting lemma generalized to unfoldings. It is shown that inhomogeneous linear state-space transformations suffice for the splitting to determine the normal forms for both singularities and unfoldings (although not the mappings). A splitting algorithm is given that may be implemented by computer algebra, and together with previous work allows an arbitrary elementary catastrophe to be reduced to normal form.

We propose in this paper a significant refinement of normal forms for differentiable maps near a fixed point. We give a method to obtain further reduction of classical normal forms with concrete and interesting applications. Our method leads to unique normal forms either with respect to general diffeomorphisms in certain cases or with respect to near identity diffeomorphisms in other cases. Our approach is rational in the sense that if the coefficients of a map are in a field K, so is its normal form.

The present paper is a survey of some results concerning higher-order elliptic differential operators which degenerate on the boundary of a domain. The principal aspect in the study of such operators is that of investigating the corresponding ordinary equations with parameters which degenerate at a single point. The parametrix for such operators is a degenerating pseudo-differential operator. We investigate the properties of the kernels of these pseudo-differential operators.