10 research outputs found
Bifurcations and Chaos in a Periodic Predator-Prey Model
The model most often used by ecologists to describe interactions between predator and prey populations is analyzed in this paper with reference to the case of periodically varying parameters. A complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique. The results perfectly agree with the local theory of periodically forced Hopf bifurcation. The two classical routes to chaos, i.e., cascade of period doublings and torus destruction, are numerically detected
Ambiguities in input-output behavior of driven nonlinear systems close to bifurcation
Since the so-called Hopf-type amplifier has become an established element in the modeling of the mammalian hearing organ, it also gets attention in the design of nonlinear amplifiers for technical applications. Due to its pure sinusoidal response to a sinusoidal input signal, the amplifier based on the normal form of the Andronov-Hopf bifurcation is a peculiar exception of nonlinear amplifiers. This feature allows an exact mathematical formulation of the input-output characteristic and thus deeper insights of the nonlinear behavior. Aside from the Hopf-type amplifier we investigate an extension of the Hopf system with focus on ambiguities, especially the separation of solution sets, and double hysteresis behavior in the input-output characteristic. Our results are validated by a DSP implementation
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Analysis and design of nonlinear resonances via singularity theory
Bifurcation theory and continuation methods are well-established tools for the analysis of nonlinear mechanical systems subject to periodic forcing. We illustrate the added value and the complementary information provided by singularity theory with one distinguished parameter. While tracking bifurcations reveals the qualitative changes in the behaviour, tracking singularities reveals how structural changes are themselves organised in parameter space. The complementarity of that information is demonstrated in the analysis of detached resonance curves in a two-degree-of-freedom system.G. Habib would like to acknowledge the financial support of the Belgian National Science Foundation FRS-FNRS (PDR T.0007.15). The research leading to these results has received funding from the European Research Council under the Advanced ERC Grant Agreement Switchlet n.670645
An upper limit for slow-earthquake zones: self-oscillatory behavior through the Hopf bifurcation mechanism from a spring-block model under lubricated surfaces
"The complex oscillatory behavior of a spring-block model is analyzed via the Hopf bifurcation mechanism. The mathematical spring-block model includes Dieterich Ruina's friction law and Stribeck's effect. The existence of self-sustained oscillations in the transition zone where slow earthquakes are generated within the frictionally unstable region is determined. An upper limit for this region is proposed as a function of seismic parameters and frictional coefficients which are concerned with presence of fluids in the system. The importance of the characteristic length scale L, the implications of fluids, and the effects of external perturbations in the complex dynamic oscillatory behavior, as well as in the stationary solution, are take into consideration.
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Nonlinear resonance and excitability in interconnected systems
Engineering design amounts to develop components and interconnect them to obtain a desired behaviour. While in the context of equilibrium dynamics there is a well-developed theory that can account for robustness and optimality in this process, we still lack a corresponding methodology for nonequilibrium dynamics and in particular oscillatory behaviours. With the aim of fostering such a theory, this thesis studies two basic interconnections in the contexts of nonlinear resonance and excitability, two phenomena with the potential of encompassing a large number of applications.
The first interconnection is considered in the context of vibration absorption. It corresponds to coupling two Duffing oscillators, the prototypical example of nonlinear resonator. Of primary interest is the frequency response of the system, which quantifies the behaviour in presence of harmonic forces. The analysis focuses on how isolated families of solutions appear and merge with a main one. Using singularity theory it is possible to organise these solutions in the space of parameters and delimit their presence through numerical methods.
The second interconnection studied in this dissertation appears in the context of excitable circuits. Combining a fast excitable system and a slower oscillatory system that share a similar structure naturally leads to bursting. The resulting system has a slow-fast structure that can be leveraged in the analysis. The first step of this analysis is a novel slow-fast model of bistability between a rest state and a spiking attractor. Following this, the analysis moves to the complete interconnection, and in particular on how it can generate different patterns of bursting activity
A novel technique for high-resolution frequency discriminators and their application to pitch and onset detection in empirical musicology
This thesis presents and evaluates software for simultaneous, high-resolution time-frequency
discrimination. Whilst this is a problem that arises in many areas of engineering, the software here is developed to assist musicological investigations. In order to analyse musical performances, we must first know what is happening and when; that is, at what time each note begins to sound (the note onset) and what frequencies are present (the pitch). The work presented here focusses on onset detection, although the representation of data used for this task could also be used to track the pitch. A potential method of determining pitch on a sample-to-sample basis is given in the final chapter.
Extant software for onset detection uses standard signal processing techniques to search for changes in features like the spectrum or phase. These methods struggle somewhat, as they are constrained by the uncertainty principle, which states that, as time resolution is increased, frequency resolution must decrease and vice versa.
However, we can hear changes in frequency to a far greater time resolution than the uncertainty principle would suggest is possible. There is an active process in the inner ear which adds energy and enables this perceptual acuity. The mathematical expression which describes this system is known as the Hopf bifurcation.
By building a bank of tuned resonators in software, each of which operates at a Hopf bifurcation, and driving it with audio, changes in frequency can be detected in times that defy the uncertainty relation, as we are not seeking to directly measure the time-frequency features of a system, rather it is used to drive a system. Time and frequency information is then available from the internal state variables of the system.
The characteristics of this bank of resonators - called a 'DetectorBank' - are investigated thoroughly. The bandwidth of each resonator ('detector') can be as narrow as 0.922Hz and the system bandwidth is extended to the Nyquist frequency. A nonlinear system may be expected to respond poorly when presented with multiple simultaneous input frequencies; however, the DetectorBank performs well under these circumstances.
The data generated by the DetectorBank is then analysed by an OnsetDetector. Both the development and testing of this OnsetDetector are detailed. It is tested using a repository of recordings of individual notes played on a variety of instruments, with promising results. These results are discussed, problems with the current implementation are identified and potential solutions presented.
This OnsetDetector can then be combined with a PitchTracker to create a NoteDetector, capable of detecting not only a single note onset time and pitch, but information about changes that occur within a note.
Musical notes are not static entities: they contain much variation. Both the performer's intonation and the characteristics of the instrument itself have an effect on the frequency present, as well as features like vibrato. Knowledge of these frequency components, and how they appear or disappear over the course of the note, is valuable information and the software presented here enables the collection of this data