Algebraic treatment of the Pais–Uhlenbeck oscillator and its PT-variant

The algebraic method enables one to study the properties of the spectrum of a quadratic Hamiltonian through the mathematical properties of a matrix representation called regular or adjoint. This matrix exhibits exceptional points where it becomes defective and can be written in canonical Jordan form. It is shown that any quadratic function of K coordinates and K momenta leads to a 2K differential equation for those dynamical variables. We illustrate all these features of the algebraic method by means of the Pais–Uhlenbeck oscillator and its PT-variant.

Numerical Detection and Analysis of Strong Resonance Bifurcations with a Reflection Symmetry and Some Applications in Economics and Neural Networks

This paper is concerned with the strong resonance bifurcations with a reflection symmetry i.e. [Formula: see text]-symmetry in maps. We compute the normal form of [Formula: see text] resonance and [Formula: see text] resonance bifurcations with [Formula: see text]-symmetry. We use standard normal form techniques in order to obtain the reduced map. Then, we will obtain explicit formulae for normal form coefficients of bifurcations with [Formula: see text]-symmetry. By using critical coefficients, we avoid the computation of the center manifold and the transformation of the linear part of the map into Jordan form. So this method can be used in the study of bifurcations with [Formula: see text]-symmetry in general problems. To illustrate our results, we will analyze local bifurcations of the strong resonance bifurcations with [Formula: see text]-symmetry numerically and then we will present some applications from economics and neural networks.

Jordan Form-Based Algebraic Conditions for Controllability of Multiagent Systems under Directed Graphs

Based on the Jordan form of system matrix, this paper discusses algebraic conditions for the controllability of the multiagent network system with directed graph from two aspects: leader-follower network attribute and coupling input disturbance. Leader-follower network attribute refers to the topology structure and information communication among agents. Coupling input disturbance includes the number of external coupling inputs and the selection of leader nodes. When the leader-follower network attribute is fixed, the selection method of coupling input disturbance is studied for the controllability, and when the coupling input disturbance is known, we derive necessity and sufficiency conditions to determine the controllability. The reliability of theoretical results is verified by numerical examples and model simulation. Besides, the generally perfect controllability is introduced, that is, the system is always controllable regardless of the number and the locations of leaders. In practical engineering applications, the perfectly controllable topology can improve the system fault tolerance and accelerate the commercialization process, which has a profound significance for promoting the modernization process.

Drazin Inverse of Jordan Form Matrix

One type of Generalized Inverse is Drazin Inverse, or some people say the spectral inverse.In this note, we pointed out that through Jordan Form Matrices, the Drazin Inverse is well understood. \\{\tt The authors would like dedicated this note to Prof. Irawati and Prof. Sri Wahyuni on their 60th Birthday.}