Differentiable lyapunov function and center manifold theory

2007 ◽  
pp. 143-148 ◽  
Author(s):  
Frédéric Mazenc
Keyword(s):  
Lyapunov Function ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Manifold Theory

Application of Center Manifold Theory to Regulation of a Flexible Beam

1997 ◽  
Vol 119 (2) ◽  
pp. 158-165 ◽  
Author(s):  
Amir Khajepour ◽  
M. Farid Golnaraghi ◽  
Kirsten A. Morris
Keyword(s):  
Control Strategy ◽  
Center Manifold ◽  
Flexible Beam ◽  
Trial And Error ◽  
Center Manifold Theory ◽  
Control Techniques ◽  
Lumped Parameter ◽  
The Stability ◽  
Frequency Relationship ◽  
Manifold Theory

In this paper we consider the problem of regulation of a flexible lumped parameter beam. The controller is an active/passive mass-spring-dashpot mechanism which is free to slide along the beam. In this problem the plant/controller equations are coupled and nonlinear, and the linearized equations of the system have two uncontrollable modes associated with a pair of pure imaginary eigenvalues. As a result, linear control techniques as well as most conventional nonlinear control techniques can not be applied. In earlier studies Golnaraghi (1991) and Golnaraghi et al. (1994) a control strategy based on Internal resonance was developed to transfer the oscillatory energy from the beam to the slider, where it was dissipated through controller damping. Although these studies provided very good understanding of the control strategy, the analytical method was based on perturbation techniques and had many limitations. Most of the work was based on numerical techniques and trial and error. In this paper we use center manifold theory to address the shortcomings of the previous studies, and extend the work to a more general control law. The technique is based on reducing the dimension of system and simplifying the nonlinearities using center manifold and normal forms techniques, respectively. The simplified equations are used to investigate the stability and to develop a relation for the optimal controller/plant natural frequencies at which the maximum transfer of energy occurs. One of the main contributions of this work is the elimination of the trial and error and inclusion of damping in the optimal frequency relationship.

Twelve Limit Cycles in 3D Quadratic Vector Fields with Z3 Symmetry

2018 ◽  
Vol 28 (11) ◽  
pp. 1850139 ◽  
Author(s):  
Laigang Guo ◽  
Pei Yu ◽  
Yufu Chen
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Center Manifold ◽  
Three Dimensional ◽  
Center Manifold Theory ◽  
Normal Form Theory ◽  
Quadratic Vector Fields ◽  
Fine Focus ◽  
Form Theory ◽  
Manifold Theory

This paper is concerned with the number of limit cycles bifurcating in three-dimensional quadratic vector fields with [Formula: see text] symmetry. The system under consideration has three fine focus points which are symmetric about the [Formula: see text]-axis. Center manifold theory and normal form theory are applied to prove the existence of 12 limit cycles with [Formula: see text]–[Formula: see text]–[Formula: see text] distribution in the neighborhood of three singular points. This is a new lower bound on the number of limit cycles in three-dimensional quadratic systems.

Stability and Hopf Bifurcation of a Delayed Mutualistic System

2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Ruimin Zhang ◽  
Xiaohui Liu ◽  
Chunjin Wei
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Stage Structure ◽  
Center Manifold Theory ◽  
Mutualistic Relationship ◽  
Normal Form Method ◽  
Explicit Formulae ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

In this paper, we study a classic mutualistic relationship between the leaf cutter ants and their fungus garden, establishing a time delay mutualistic system with stage structure. We investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of the associated characteristic equation. By means of the center manifold theory and normal form method, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Dynamical system analysis of a nonminimally coupled scalar field

2019 ◽  
Vol 28 (15) ◽  
pp. 1950173 ◽  
Author(s):  
Subhajyoti Pal ◽  
Sudip Mishra ◽  
Subenoy Chakraborty
Keyword(s):  
Scalar Field ◽  
Critical Points ◽  
Center Manifold ◽  
System Analysis ◽  
Nonlinear Differential Equations ◽  
Center Manifold Theory ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Flat Spacetime ◽  
Manifold Theory

This paper deals with a nonminimally coupled scalar field in the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) flat spacetime. As Einstein field equations are coupled second-order nonlinear differential equations, it is very hard to find exact solutions. By suitable choice of variables, we transform Einstein field equations to an autonomous system and critical points are determined. We use center manifold theory to characterize nonhyperbolic critical points and are found to be saddle in nature. We discuss possible bifurcation scenarios, which indicate the existence of the cosmological bouncing model.

Center Manifold Theory in Infinite Dimensions

1992 ◽  
pp. 125-163 ◽  
Author(s):  
A. Vanderbauwhede ◽  
G. Iooss
Keyword(s):  
Center Manifold ◽  
Center Manifold Theory ◽  
Infinite Dimensions ◽  
Manifold Theory

scholarly journals Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xinhong Pan ◽  
Min Zhao ◽  
Chuanjun Dai ◽  
Yapei Wang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Delay Effect ◽  
Delay Differential System ◽  
The Stability ◽  
Delay Differential ◽  
Manifold Theory ◽  
Two Equilibria

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

Order Reduction of Nonlinear Time Periodic Systems Using Invariant Manifolds

2003 ◽  
Author(s):  
S. C. Sinha ◽  
Sangram Redkar ◽  
Eric A. Butcher ◽  
Venkatesh Deshmukh
Keyword(s):  
Center Manifold ◽  
Invariant Manifolds ◽  
Basic Problem ◽  
Equations Of Motion ◽  
Order Reduction ◽  
Center Manifold Theory ◽  
Manifold Theory ◽  
Time Invariant ◽  
Time Periodic ◽  
Linear And Nonlinear Systems

The basic problem of order reduction of linear and nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via Time Periodic Center Manifold Theory. A ‘reducibility condition’ is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show applications to real problems. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘combination’ resonances are discussed.

scholarly journals Bifurcation and Patterns Analysis for a Spatiotemporal Discrete Gierer-Meinhardt System

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 243
Author(s):  
Biao Liu ◽  
Ranchao Wu
Keyword(s):  
Bifurcation Theory ◽  
Center Manifold ◽  
Turing Instability ◽  
Flip Bifurcation ◽  
Center Manifold Theory ◽  
Pattern Dynamics ◽  
Instability Theory ◽  
Pattern Formations ◽  
Manifold Theory ◽  
Theoretical Results

The Gierer-Meinhardt system is one of the prototypical pattern formation models. The bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system are investigated via the couple map lattice model (CML) method in this paper. The linear stability of the fixed points to such spatiotemporal discrete system is analyzed by stability theory. By using the bifurcation theory, the center manifold theory and the Turing instability theory, the Turing instability conditions in flip bifurcation and Neimark–Sacker bifurcation are considered, respectively. To illustrate the above theoretical results, numerical simulations are carried out, such as bifurcation diagram, maximum Lyapunov exponents, phase orbits, and pattern formations.

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