Analysis of stability and bifurcations of limit cycles in Chua's circuit through the harmonic-balance approach

Abstract

This paper presents a spectral approach, based on the harmonic-balance technique, for detecting limit-cycle bifurcations in complex nonlinear circuits. The key step of the proposed approach is a method for a simple and effective computation of the Floquet multipliers (FM's) that yield stability and bifurcation conditions. As a case-study, a quite complex system, Chua's circuit, is considered. It is shown that the spectral approach is able to accurately evaluate the most significant bifurcation curves.

Stability Analysis of Power Amplifiers Under Output Mismatch Effects

“…8(d) shows the pole evolution in the complex plane. The two pairs of complex-conjugate poles f in -f and f in + f (associated with the same Floquet multiplier [11,24]) cross the imaginary axis at P in = -8.71 dBm. At the crossing point, the frequency of the poles is f in -f = 780.7 MHz [ Fig.…”

“…The results have been validated using various passive loads, which have enabled a comparison with well-established analysis techniques, such as pole-zero identification [21][22][23][24][25] or bifurcation detection with auxiliary generators [11,14,26]. Measurements have also been carried out for the two amplifiers, which exhibited unstable behavior under the conditions predicted by the method.…”

“…This is different from regular stability analyses that require knowledge of the frequency response of all the circuit elements [17][18][19][20][21][22][23][24][25]. Two different situations are analyzed in two nonlinear amplifiers used as demonstrators.…”

“…Equation (24) provides the set of Hopf bifurcation points in the  l plane, arranged in a manner different from the Hopf loci obtained with (16). Limiting the  u values considered in (24) to those providing negative resistance at the lower sideband frequency will enable a substantial reduction in computational cost compared with the exhaustive systematic analysis in Section III.…”

Rotary Traveling-Wave Oscillator With Differential Nonlinear Transmission Lines

“…At each Hopf bifurcation, a periodic solution is generated. The stability of periodic solutions is determined by a set of L Floquet multipliers m1, m2 to mL, where L is the system dimension [37][38]. A stable periodic oscillation must have a real multiplier with value 1 (m1=1), associated with the system autonomy, and the rest of its real and complex-conjugat e multipliers must have magnitude smaller than one, that is, |mi|<1, where i=2 … L-1.…”