Degenerate bifurcations and catastrophe sets via frequency analysis

We present some bifurcation conditions using the well-known stability analysis of feedback systems. A general ordinary differential equation system is formulated in two parts: one that considers the linear part and the other that includes the memoryless nonlinear part, in a similar way as the describing function. The bifurcation conditions are obtained using the results of the generalized Nyquist stability criterion (GNSC) with some explicit formulae derived from some properties of the complex variable

We analyse simultaneously both static and dynamic (Hopf) bifurcations and their degeneracies in a rich example, a continuous stirred-tank reactor (CSTR), in which two consecutive, irreversible, first-order reactions A→B→C occur     

Influence of period‐doubling bifurcations in the appearance of border collisions for a ZAD‐strategy‐controlled buck converter

The first period‐doubling bifurcation of a dc–dc buck converter controlled by a zero‐average dynamic strategy is studied in detail. Owing to the saturation of the duty cycle, this bifurcation is followed by a border‐collision bifurcation, which is the main mechanism to introduce instability and chaos in the circuit. The multiparameter analysis presented here leads to a complete knowledge of the relatioship between these two bifurcations. The results are obtained by using a frequency‐domain approach for the study of period‐two oscillations in maps. Copyright © 2010 John Wiley & Sons, Ltd.     

A Frequency Domain Method for Analyzing Period Doubling Bifurcations in Discrete-Time Systems

A frequency domain methodology for the approximation of period doubling bifurcations in discrete-time nonlinear systems is presented. The computation of a stability index, which characterizes the dynamical behavior of the emerging period-two orbit is also derived. The technique is applied to estimate the domain of attraction of the fixed point in an adaptive control system and to improve the dynamical behavior of a buck converter.     

IL-12 is involved in the induction of experimental autoimmune myasthenia gravis, an antibody-mediated disease

IL-12 has been shown to be involved in the pathogenesis of Th1-mediated autoimmune diseases, but its role in antibody-mediated autoimmune pathologies is still unclear. We investigated the effects of exogenous and endogenous IL-12 in experimental autoimmune myasthenia gravis (EAMG). EAMG is an animal model for myasthenia gravis, a T cell-dependent, autoantibody-mediated disorder of neuromuscular transmission caused by antibodies to the muscle nicotinic acetylcholine receptor (AChR). Administration of IL-12 with Torpedo AChR (ToAChR) to C57BL/6 (B6) mice resulted in increased ToAChR-specific IFN-gamma production and increased anti-ToAChR IgG2a serum antibodies compared with B6 mice primed with ToAChR alone. These changes were associated with earlier and greater neurophysiological evidence of EAMG in the IL-12-treated mice, and reduced numbers of AChR. By contrast, when IL-12-deficient mice were immunized with ToAChR, ToAChR-specific Th1 cells and anti-ToAChR IgG2a serum antibodies were reduced compared to ToAChR-primed normal B6 mice, and the IL-12-deficient mice showed almost no neurophysiological evidence of EAMG and less reduction in AChR. These results indicate an important role of IL-12 in the induction of an antibody-mediated autoimmune disease, suggest that Th1-dependent complement-fixing IgG2a anti-AChR antibodies are involved in the pathogenesis of EAMG, and help to account for the lack of correlation between anti-AChR levels and clinical disease seen in many earlier studies.     

Computations of limit cycles via higher-order harmonic balanceapproximation

The detection of limit cycles arising from Hopf bifurcation phenomena by applying the harmonic balance method with different higher-order approximations is discussed. The results are presented using a graphical procedure that indicates clearly how the predictions of amplitude and frequency of a periodic solution can be improved by using higher and higher order approximations. Complete and explicit formulas for eighth-order harmonic balance approximation are provided     

Computing bifurcation points via characteristics gain loci

The authors show how to check the crossing on the imaginary axis by the eigenvalues of the linearized system of differential equations depending on a real parameter μ via feedback system theory. E. Hopf's theorem (1942) refers to a system of ordinary differential equations depending on the real parameter μ in which, when a single pair of complex conjugate eigenvalues of the linearized equations cross the imaginary axis under the parameter vibration, near this critical condition periodic orbits appear. The authors present simple formulas for both static (one eigenvalue zero) and dynamic or Hopf (a single pure imaginary pair) bifurcations, and show some singular conditions (degeneracies) by continuing the bifurcation curves in the steady-state manifold. The bifurcation curves and singular sets of an interesting chemical reactor which possesses multiplicity in the equilibrium solutions and in the Hopf bifurcation points are described