Javier Ros

LIMIT CYCLE BIFURCATION FROM CENTER IN SYMMETRIC PIECEWISE-LINEAR SYSTEMS

The rapid bifurcation described by Kriegsmann [1987] is shown to be a generic bifurcation for planar symmetric piecewise-linear systems. The bifurcation can be responsible for the abrupt appearance of stable periodic oscillations. Although it has some similarities with the Hopf bifurcation for smooth systems, since the stability change of an equilibrium involves the appearance of one limit cycle, the dependence of the limit cycle amplitude on the bifurcation parameter is different from the Hopfs case. To characterize this bifurcation, accurate estimates for the amplitude and period of the bifurcating limit cycle are given. The analysis is just illustrated with the application of the theoretical results to the Wien bridge oscillator. Comparisons with experimental data and Kriegsmanns analysis are also included.

LIMIT CYCLE BIFURCATION IN 3D CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO ZONES: APPLICATION TO CHUA'S CIRCUIT

The generic case of three-dimensional continuous piecewise linear systems with two zones is analyzed. From a bounded linear center configuration we prove that the periodic orbit which is tangent to the separation plane becomes a limit cycle under generic conditions. Expressions for the amplitude, period and characteristic multipliers of the bifurcating limit cycle are given. The obtained results are applied to the study of the onset of asymmetric periodic oscillations in Chuas oscillator.

Limit Cycle and Boundary Equilibrium Bifurcations in Continuous Planar Piecewise Linear Systems

Boundary equilibrium bifurcations in continuous planar piecewise linear systems with two and three zones are considered, with emphasis on the possible simultaneous appearance of limit cycles. Situations with two limit cycles surrounding the only equilibrium point are detected and rigorously shown for the first time in the family of systems under study. The theoretical results are applied to the analysis of an electronic Wien bridge oscillator with biased polarization, characterizing the different parameter regions of oscillation.

A BIPARAMETRIC BIFURCATION IN 3D CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO ZONES: APPLICATION TO CHUA'S CIRCUIT

In this paper, a possible degeneration of the focus-center-limit cycle bifurcation for piecewise smooth continuous systems is analyzed. The case of continuous piecewise linear systems with two zones is considered, and the coexistence of two limit cycles for certain values of parameters is justified. Finally, the Chuas circuit is shown to exhibit the analyzed bifurcation. The obtained bifurcation set in the parameter plane is similar to the degenerate Hopf bifurcation for differentiable systems.

Teaching Mechanism and Machine Theory with GeoGebra

Teaching Mechanism and Machine Theory involves the description of mechanisms which are not easy to represent with static drawings. Computer applications are a common way to reproduce the motion of mechanisms. GeoGebra is a free dynamic geometry software with interactive graphics that can help in this task. By introducing mathematical constraints, such as constant distances, intersections, tangency or perpendicularity, it is possible to build a wide range of interactive mechanisms. This software has been used at the Public University of Navarra on a Degree in Mechanical Engineering, having a successful experience with the students. The software has been used in two ways. On the one hand, some mechanisms have been prepared and shown to the students in theoretical lessons in order to explain their motion. On the other hand, practical lessons have been carried out, in which the students have programmed one degree of freedom mechanisms with GeoGebra.Students have shown interest in the practical lessons and results have been satisfactory.

On the fold-Hopf bifurcation for continuous piecewise linear differential systems with symmetry

In this paper a partial unfolding for an analog to the fold-Hopf bifurcation in three-dimensional symmetric piecewise linear differential systems is obtained. A particular biparametric family of such systems is studied starting from a very degenerate configuration of nonhyperbolic periodic orbits and looking for the possible bifurcation of limit cycles. It is proved that four limit cycles can coexist after perturbation of the original configuration, and other two limit cycles are conjectured. It is shown that the described bifurcation scenario appears for appropriate values of parameters in the celebrated Chuas oscillator.

The Focus-Center-Limit Cycle Bifurcation in Discontinuous Planar Piecewise Linear Systems Without Sliding

Planar discontinuous piecewise linear systems with two linearity zones, one of them being of focus type, are considered. By using an adequate canonical form under certain hypotheses, the bifurcation of a limit cycle, when the focus changes its stability after becoming a linear center, is completely characterized. Analytic expressions for the amplitude, period and characteristic multiplier of the bifurcating limit cycle are provided. The studied bifurcation appears in real world applications, as shown with the analysis of an electronic Wien bridge oscillator without symmetry.

ON PERIODIC ORBITS OF 3D SYMMETRIC PIECEWISE LINEAR SYSTEMS WITH REAL TRIPLE EIGENVALUES

The counter-intuitive appearance of stable periodic orbits in three-dimensional piecewise linear systems has been recently reported for stable saturated control systems with high gain and real triple eigenvalues [Moreno & Suarez 2004]. In this letter using several mathematical tools, the reported sudden phenomenon is explained, and the analysis is completed by providing a global bifurcation diagram of the symmetric periodic orbits. Approximate methods are used only to illustrate the nonlinear behavior with respect to the bifurcation parameter. Analytical methods are employed to rigorously prove the main result. The employed techniques are useful not only for the family studied but also for generic three-dimensional symmetric piecewise linear systems.

Some Recent Results for Continuous Switched Linear Systems

The elemental structure arising from the continuous autonomous switching of two linear systems is considered. After introducing certain canonical forms, some analytical results about limit cycle bifurcation are reported, showing that such systems generically exhibit a jump transition to oscillating behavior. Explicit expressions for quantitative characteristics of the periodic oscillation are obtained for the cases of dimension two and three. As another relevant result, it is shown that continuous n-dimensional switched linear systems whose both components are Hurwitz need not be globally asymptotically stable when n is greater or equal to 3.

On Discontinuous Piecewise Linear Models for Memristor Oscillators

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems. The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.