Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics

A new approach for the analysis of nonlinear microwave circuits, the named frequency-domain continuation method, is presented. It allows the unified, but nevertheless efficient, computer analysis of autonomous and forced nonlinear circuits by utilizing the globalization properties of the continuation technique, and provides stability information in parallel. Its theoretical background and relation to bifurcation theory is outlined. Its high potential is demonstrated by application to problems with single- and multiple-frequency input, strong nonlinearities, hysteresis phenomena and subharmonics.

Frequency domain continuation method for the analysis and stability investigation of nonlinear microwave circuits

We investigate the dynamics of thermoacoustic systems with multiple linearly unstable modes. If a linear analysis reveals more than one mode with positive growth rate, nonlinear methods have to be used to determine the existence and stability of steady-state oscillations. One possible way to engage this problem is a first-order harmonic balance approach based on describing function representations for the flame response. In contrast to the case of a single unstable mode, the nonlinearity output to multiple sinusoidal components with different frequencies and amplitudes has to be known. Based on this approach, we present conditions for the existence and stability of single- or multi-mode steady-state oscillations. We apply this method to a thermoacoustic model system having two linearly unstable modes. By varying one of the system parameters, we find stable and unstable single-mode steady-states as well as unstable simultaneous oscillations. Associated with the stability of the single-mode limit cycles, we...

https://scd.sagepub.com/content/4/1/1.full.pdf

Nonlinear interactions of multiple linearly unstable thermoacoustic modes

The use of the harmonic balance method for the control of period doubling bifurcations is considered. The objectives of the control design can include delay of a given period doubling bifurcation as well as stabilization. The results are inherently approximate in nature. >

Control of period doubling bifurcations using harmonic balance

We develop and discuss the generation of new limit cycles in chemical systems by external periodic perturbations. The deterministic analysis is based on prior work by Loud. We choose a model derived from experiments, the conversion of 2,3 epoxy‐1‐propanol to glycerine. The autonomous system of equations which govern the dynamics of this system has a limit cycle arising from a normal Hopf bifurcation, and a stable focus, and a limit cycle associated with an inverted Hopf bifuraction. For specific choices of periodic perturbations a prescribed number of periodic attractors can be generated from a single periodic attractor (normal Hopf bifurcation); and two periodic attractors, a periodic and biperiodic, or two biperiodic attractors can be generated from a stable focus and periodic attractor (inverted Hopf bifurcation). Noise is then imposed on the attractors of the autonomous system and the multiple attractors of the externally forced system. Noise imposed on the autonomous system with a stable focus and limit cycle causes random transitions, analogous to first order phase transitions. This behavior has some resemblance to intermittency noted in some biological systems. From the results of noise imposed on systems with different numbers of isoperiodic limit cycles we suggest, after a limited number of observations, the trend that as the number of attractors increases, the lines in the power spectrum narrow and the area in the phase plane within contours of equal probability increases.

Generation of multiple attractors and nonequilibrium phase transitions

This survey paper presents a comprehensive review of work in the field of non-linear oscillations. A brief discussion of second-order systems is followed by a presentation of exact criteria, approximate analytical methods and computational techniques for limit cycles in single variable systems. Multivariate systems are then covered from an analogous viewpoint which allows the reader to clearly identify both how single variable methods have been extended and the possibilities for further research. Particular emphasis is placed on describing function methods since it is believed that, where exact solutions are not possible, the approach may not only give approximate solutions but provides good insight for further computational or simulation studies. The coverage is essentially restricted to continuous lumped parameter systems and includes both free and forced oscillations. Several applications of the theories in various fields of engineering and science are discussed and indicate the broad interest in non-l...

H. T. Dorrah

Papers

A survey on non-linear oscillations