Fernando Verduzco

ANALYSIS OF THE TAKENS-BOGDANOV BIFURCATION ON m-PARAMETERIZED VECTOR FIELDS

Given an m-parameterized family of n-dimensional vector fields, such that: (i) for some value of the parameters, the family has an equilibrium point, (ii) its linearization has a double zero eigenvalue and no other eigenvalue on the imaginary axis, sufficient conditions on the vector field are given such that the dynamics on the two-dimensional center manifold is locally topologically equivalent to the versal deformation of the planar Takens–Bogdanov bifurcation.

The Boundary Crossing Theorem and the Maximal Stability Interval

The boundary crossing theorem and the zero exclusion principle are very useful tools in the study of the stability of family of polynomials. Although both of these theorem seem intuitively obvious, they can be used for proving important results. In this paper, we give generalizations of these two theorems and we apply such generalizations for finding the maximal stability interval.

Bifurcations and Chaos in a PD-Controlled Pendulum

The complex dynamics of a pendulum controlled by a Proportional-Derivative (PD) compensator are analyzed. A classification of equilibrium points and the characterization of their bifurcations is also presented. It is shown that the controlled pendulum may exhibit a chaotic behavior when the desired position is periodic and the proportional gain and total dissipation are small enough.

BIFURCATION ANALYSIS OF A 2-DOF ROBOT MANIPULATOR DRIVEN BY CONSTANT TORQUES

The bifurcations of a two-degree-of-freedom (2-DOF) robot manipulator with linear viscous damping and constant torques at joints are analyzed at three equilibrium points. We show that, provided some conditions on the parameters are satisfied, two of these equilibria have a Jacobian matrix with a double zero eigenvalue and a pair of pure imaginary eigenvalues, while the other one has a quadruple zero eigenvalue. We use the center manifold theorem and the normal form theory to show the presence of different kinds of local bifurcations, ranging from codimension one (Hopf and fold), three (cusp and degenerate zero-Hopf), and higher (double zero, double zero-Hopf, triple zero, quadruple zero and Hopf–Hopf).

Control of the Hopf Bifurcation by a Linear Feedback Control

In this work, we design a kind of linear state feedback control, via the roots connecting-curve, for a class of nonlinear systems which permits to control the Hopf bifurcation. An illustrative example is given.

CONTROL OF CODIMENSION ONE STATIONARY BIFURCATIONS

The control of the saddle-node, transcritical and pitchfork bifurcations are analyzed in nonlinear control systems with one zero eigenvalue. It is shown that, provided some conditions on the vector fields are satisfied, it is possible to design a control law such that the bifurcation properties can be modified in some desirable way. To simplify the analysis to dimension one, the center manifold theory is used.

Normal Form and Control of the Hopf bifurcation

A new normal form in nonlinear control systems is introduced to control the Hopf bifurcation. Very simple expressions for the so-called stability coefficients of the Hopf bifurcation are obtained through the design of simple control laws.

Homoclinic Chaos in 2-DOF Robot Manipulators Driven by PD Controllers

The existence of Smale horseshoes in the dynamics of twodegrees of freedom (2-DOF) robot manipulators with viscous damping anddriven by classical proportional-derivative (PD) controllers is proved.The situation where an actuator failure occurs is also analyzed. Thecontrollers correspond to the classical PD and the PD with gravitycompensation. They are considered as non-Hamiltonian perturbations of anundriven 2-DOF robot; then a technique developed by Holmes and Marsden,which uses a combination of a reduction scheme and Melnikovs methodwith an energy balance argument, is applied.

Stability and Bifurcations of an Underactuated Robot Manipulator 2

Abstract The stability and bifurcations of a two-degree-of-freedom (2-DOF) under-actuated robot manipulator with linear viscous damping and constant torque are studied. It is shown that, under some conditions on the parameters, the Jacobian matrix has a double zero eigenvalue and a pair of pure imaginary eigenvalues. Around such parameters, the system has a bifurcation zoo: since codimension one (Hopf and fold), three (cusp and degenerate zero-Hopf), and higher (degenerated double zeroHopf and Hopf-Hopf). The center manifold theorem and normal form theory are used, and some numerical experiments are presented to illustrate the results.

Control of the saddle-node and transcritical bifurcations

Abstract In this paper, the control of the saddle-node and transcritical bifurcations in nonlinear systems is treated. A new approach is presented to find sufficient conditions in terms of the original vector fields. The analysis of the system dynamics is reduced to dimension one through the center manifold theorem.