Experimental Bifurcation Analysis Research Articles

Parameter estimation of two coupled oscillator model for pure intrinsic thermo-acoustic instability

A nonlinear phenomenological model of two coupled oscillators is proposed, which is able to describe the rich nonlinear behaviour stemming from an unstable pure intrinsic thermo-acoustic (ITA) mode of a simple combustion system. In an experimental bifurcation analysis of a pure ITA mode, it was observed that for increasing mean upstream velocity the flames loose stability through a supercritical Hopf bifurcation and subsequently exhibit limit cycle, quasi-periodic and period-2 limit cycle oscillations. The quasi-periodic oscillations were characterised by low frequent amplitude and frequency modulation. It is shown that a phenomenological model consisting of two coupled oscillators is able to reproduce qualitatively all the different experimentally observed regimes. This model consists of a nonlinear Van der Pol oscillator and a linear damped oscillator, which are nonlinearly coupled to each other. Furthermore, a parameter estimation of the model parameters is conducted, which reveals a good quantitative match between the model response and the experimental data. Hence, the presented phenomenological dynamical model accurately describes the nonlinear self-excited acoustic behaviour of premixed flames and provides a promising model structure for nonlinear time-domain flame models.

Nonlinear dynamics of pure intrinsic thermo-acoustic modes

In this paper the nonlinear dynamics of the pure Intrinsic Thermo-Acoustic (ITA) modes are experimentally investigated for burner stabilized premixed flames by decoupling them from the acoustic modes. Thereto, a setup is used with up- and downstream mufflers that provide close to anechoic boundary conditions. This enables the investigation of the self-excited oscillations of the flame, without the influences of acoustic reflections. An experimental bifurcation analysis is conducted to systematically investigate the different asymptotic oscillatory states and their transitions. By increasing the upstream velocity, the flame looses stability through a supercritical Hopf bifurcation and reveals limit cycles with a single dominant frequency. For increasing upstream velocity, a low frequent mode appears, resulting in quasi-periodic oscillations. This regime is characterized by a beating phenomenon with simultaneous amplitude and frequency modulations. When the upstream velocity is even further increased, a period doubling bifurcation is observed. To our best knowledge, the experimental investigation of the nonlinear dynamical behavior of pure ITA modes and, in particular, the low frequency oscillation, has not been reported before, and it provides valuable insights towards developing accurate time-domain models of the nonlinear acoustic behavior of premixed flames. Finally, a phenomenological model of two coupled oscillators is introduced that is able to qualitatively reproduce all observed nonlinear oscillations of the flames.

Parameter identification of two coupled oscillator model for pure intrinsic thermoacoustic instability

Thermo-acoustic instabilities are a well-known problem in combustions systems. In general, the type of these modes can be divided in two categories: (i) modes with an acoustic origin and (ii) modes with an intrinsic thermo-acoustic (ITA) origin. It is an challenging task to observe the ITA modes experimentally. Therefore, a setup with a burner in an acoustic embedding with close to anechoic up- and downstream terminations is required. In previous research we introduced such a setup and conducted an experimental bifurcation analysis on the dominant pure ITA mode. In this bifurcation analysis, we observed that for increasing upstream velocity the flames loose stability through a supercritical Hopf bifurcation and subsequently exhibit limit cycle, quasi-periodic and period-2 limit cycle oscillations. The quasi-periodic oscillations were characterized by low frequency amplitude and frequency modulation. In this research we present a phenomenological model consisting of two coupled oscillators that is able to reproduce all the different experimentally observed regimes. This model consists of a nonlinear Van der Pol oscillator and a linear mass-spring-damper oscillator, which are nonlinearly coupled to each other. Furthermore, we executed a parameter identification of the model and concluded that the model is able to describe the experimental data quantitatively well.

Using scientific machine learning for experimental bifurcation analysis of dynamic systems

Augmenting mechanistic ordinary differential equation (ODE) models with machine-learnable structures is a novel approach to create highly accurate, low-dimensional models of engineering systems incorporating both expert knowledge and reality through measurement data. Our exploratory study focuses on training universal differential equation (UDE) models for physical nonlinear dynamical systems with limit cycles: an aerofoil undergoing flutter oscillations and an electrodynamic nonlinear oscillator. We consider examples where training data is generated by numerical simulations, whereas we also employ the proposed modelling concept to physical experiments allowing us to investigate problems with a wide range of complexity. To collect the training data, the method of control-based continuation is used as it captures not just the stable but also the unstable limit cycles of the observed system. This feature makes it possible to extract more information about the observed system than the open-loop approach (surveying the steady state response by parameter sweeps without using control) would allow. We use both neural networks and Gaussian processes as universal approximators alongside the mechanistic models to give a critical assessment of the accuracy and robustness of the UDE modelling approach. We also highlight the potential issues one may run into during the training procedure indicating the limits of the current modelling framework.

Mathematical modelling, numerical and experimental analysis of one-degree-of-freedom oscillator with Duffing-type stiffness

The aim of the work is to develop and test simple mathematical models suitable for fast and realistic simulations of bifurcation dynamics of systems with a double potential well generated by a pair of repulsive magnets positioned transversely to the direction of motion, where there are motion resistances typical for rolling guides found in industry. Mathematical modelling of a harmonically forced one-degree-of-freedom oscillator with magnetically modified elasticity generating a double symmetrical minimum of potential was carried out. The system is composed of a cart moving along a linear rolling bearing with inertial excitation by means of a stepper motor with an unbalanced disc. The stiffness is realized by a pair of neodymium magnets of axes perpendicular to the direction of motion, in a position corresponding to the static equilibrium position of the oscillator, with additional mechanical linear springs, which generate a system similar to the Duffing system. Different models of the interaction force between cylindrical neodymium magnets were developed. The model parameters are estimated based on experimental data. Then the model is validated through additional experimental and numerical bifurcation analysis of the system. A very good agreement between numerical simulations and experimental data is obtained.

A nonlinear frequency-dependent spring-mass model for estimating loading caused by rhythmic human jumping

An empirical nonlinear, frequency-dependant, spring-mass system is conjectured for modelling human rhythmic jumping. This model is vital for correctly estimating human-structure dynamic interactions. An experimental study was employed to evaluate the leg mechanics and dynamic loading of a human jumper. Testing was performed over a large range of prescribed jumping frequencies. Subjects performed rhythmic jumps on a force plate and they were monitored by a motion capture system from which the displacement of the centre of mass was identified. Least squares system identification was utilised to determine the parameters of the spring-mass model for human rhythmic jumping. A nonlinear stiffness, rather than a conventional linear spring, is proposed to better capture the observed behaviour during periodic jumping. Force-displacement curves of each subject, during the contact phase of rhythmic jumping, were explored. These display an array of both classical Duffing’s type nonlinear softening and hardening spring stiffnesses over the range of jumping frequencies. The coefficients of the Duffing’s type model are observed to be highly sensitive to jumping frequency. A Poincaré section (phase-space) representation is used to visualise the jumping attractor’s topology. Thus, an experimental bifurcation analysis is performed suggesting the presence of both period doubling and fold bifurcations. These describe the transition from observed period-2 to period-1 jumping and coexisting low/high amplitude jumping behaviour. This study presents a framework for characterising the nonlinear loading of a human performing rhythmic jumping from direct measurements of force and displacement.

Theoretical and experimental bifurcation analysis of rock under plane strain compression

A new plane strain apparatus for rock testing was designed and built, and a series of plane strain tests on sandstone was performed. The formation and orientation of localization in sandstone were studied theoretically within the framework of equilibrium bifurcation theory. Corresponding elastoplastic constitutive models with nonlinear yield and potential functions were developed and fitted on the plane strain test data. The inclination of the shear band and the bifurcation point were predicted using the acoustic tensor method. The comparison of the computational and experimental results indicates that the initiation of a shear band predicted by a two-dimensional model could occur within the strain hardening region (i.e. prior to the peak stress), and the predicted orientations of the shear bands agreed qualitatively with those measured.

Experiment-Based Motion Reconstruction and Restitution Coefficient Estimation of a Vibro-Impact System

The objective of this paper is to demonstrate the motion reconstruction and the parameter estimation of a vibro-impact (VI) system from limited experimental information. Based on the measured displacement and acceleration of its linear main system, the rest motion information such as the displacement and velocity of the attached VI energy sink can be calculated rather than difficult direct measurement, and therefore, different response regimes from the strongly modulated response to the classic regime with two impacts per cycle are reconstructed. Consequently, it provides comprehensive experimental data for the validation of analytical and numerical results and for any experimental bifurcation analysis. Moreover, a procedure to estimate the restitution coefficient from periodic impacts is demonstrated. This new experimental approach to estimate the value of the restitution coefficient is simple and this accurate value could play an important role in analytical and numerical study.

A Practice-Oriented Bifurcation Analysis for Pulse Energy Converters. Part 2: An Operating Regime

The paper continues the discussion on bifurcation analysis for applications in practice-oriented solutions for pulse energy conversion systems (PEC-systems). Since a PEC-system represents a nonlinear object with a variable structure, then the description of its dynamics evolution involves bifurcation analysis conceptions. This means the necessity to resolve the conflict-of-units between the notions used to describe natural evolution (i.e. evolution of the operating process towards nonoperating processes and vice versa) and the notions used to describe a desirable artificial regime (i.e. an operating regime). We consider cause-effect relations in the following sequence: nonlinear dynamics-output signal-operating characteristics, where these characteristics include stability and performance. Then regularities of nonlinear dynamics should be translated into regularities of the output signal dynamics, and, after, into an evolutional picture of each operating characteristic. In order to make the translation without losses, we first take into account heterogeneous properties within the structures of the operating process in the parametrical (P-) and phase (X-) spaces, and analyze regularities of the operating stability and performance on the common basis by use of the modified bifurcation diagrams built in joint PX-space. Then, the correspondence between causes (degradation of the operating process stability) and effects (changes of the operating characteristics) is decomposed into three groups of abnormalities: conditionally unavoidable abnormalities (CU-abnormalities); conditionally probable abnormalities (CP-abnormalities); conditionally regular abnormalities (CR-abnormalities). Within each of these groups the evolutional homogeneity is retained. After, the resultant evolution of each operating characteristic is naturally aggregated through the superposition of cause-effect relations in accordance with each of the abnormalities. We demonstrate that the practice-oriented bifurcation analysis has fundamentally specific purposes and tools, like for the computer-based bifurcation analysis and the experimental bifurcation analysis. That is why, from our viewpoint, it seems to be a rather novel direction in the general context of bifurcation analysis conceptions. We believe that the discussion could be interesting to pioneer research intended for the design of promising systems of pulse energy conversion.

Guess-Work and Reasonings on Centennial Evolution of Surface Air Temperature in Russia. Part IV: Towards Economic Estimations of Climate-Related Damages from the Bifurcation Analysis Viewpoint

The paper completes the cycle of the research devoted to the development of the experimental bifurcation analysis (not computer simulations) in order to answer the following questions: whether qualitative changes occur in the dynamics of local climate systems in a centennial timescale?; how to analyze such qualitative changes with daily resolution for local and regional space-scales?; how to establish one-to-one daily correspondence between the dynamics evolution and economic consequences for productions? To answer the questions, the unconventional conceptual model to describe the local climate dynamics was proposed and verified in the previous parts. That model (HDS-model) originates from the hysteresis regulator with double synchronization and has a variable structure due to competition between the amplitude quantization and the time quantization. The main advantage of the HDS-model is connected with the possibility to describe “internally” (on the basis of the self-regulation) the specific causal effects observed in the dynamics of local climate systems instead of “external” description of three states of the hysteresis behavior of climate systems (upper, lower and transient states). As a result, the evolution of the local climate dynamics is based on the bifurcation diagrams built by processing the data of meteorological observations, where the strange effects of the essential interannual daily variability of annual temperature variation are taken into account and explained. It opens the novel possibilities to analyze the local climate dynamics taking into account the observed resultant of all internal and external influences on each local climate system. In particular, the paper presents the viewpoint on how to estimate economic damages caused by climate-related hazards through the bifurcation analysis. That viewpoint includes the following ideas: practically each local climate system is characterized by its own time pattern of the natural qualitative changes in temperature dynamics over a century, so, any unified time window to determine the local climatic norms seems to be questionable; the temperature limits determined for climate-related technological hazards should be reasoned by the conditions of artificial human activity, but not by the climatic norms; the damages caused by such hazards can be approximately estimated in relation to the average annual profit of each production. Now, it becomes possible to estimate the minimal and maximal numbers of the specified hazards per year in order, first of all, to avoid unforeseen latent damages. Also, it becomes possible to make some useful relative estimation concerning damage and profit. We believe that the results presented in the cycle illustrate great practical competence of the current advances in the experimental bifurcation analysis. In particular, the developed QHS-analysis provides the novel prospects towards both how to adapt production to climatic changes and how to compensate negative technological impacts on environment.

Guess-Work and Reasonings on Centennial Evolution of Surface Air Temperature in Russia. Part III: Where is the Joint Between Norms and Hazards from a Bifurcation Analysis Viewpoint?

The paper continues the application of the bifurcation analysis in the research on local climate dynamics based on processing the historically observed data on the daily average land surface air temperature. Since the analyzed data are from instrumental measurements, we are doing the experimental bifurcation analysis. In particular, we focus on the discussion where is the joint between the normal dynamics of local climate systems (norms) and situations with the potential to create damages (hazards)? We illustrate that, perhaps, the criteria for hazards (or violent and unfavorable weather factors) relate mainly to empirical considerations from human opinion, but not to the natural qualitative changes of climate dynamics. To build the bifurcation diagrams, we base on the unconventional conceptual model (HDS-model) which originates from the hysteresis regulator with double synchronization. The HDS-model is characterized by a variable structure with the competition between the amplitude quantization and the time quantization. Then the intermittency between three periodical processes is considered as the typical behavior of local climate systems instead of both chaos and quasi-periodicity in order to excuse the variety of local climate dynamics. From the known specific regularities of the HDS-model dynamics, we try to find a way to decompose the local behaviors into homogeneous units within the time sections with homogeneous dynamics. Here, we present the first results of such decomposition, where the quasi-homogeneous sections (QHS) are determined on the basis of the modified bifurcation diagrams, and the units are reconstructed within the limits connected with the problem of shape defects. Nevertheless, the proposed analysis of the local climate dynamics (QHS-analysis) allows to exhibit how the comparatively modest temperature differences between the mentioned units in an annual scale can step-by-step expand into the great temperature differences of the daily variability at a centennial scale. Then the norms and the hazards relate to the fundamentally different viewpoints, where the time sections of months and, especially, seasons distort the causal effects of natural dynamical processes. The specific circumstances to realize the qualitative changes of the local climate dynamics are summarized by the notion of a likely periodicity. That, in particular, allows to explain why [Formula: see text]-year averaging remains the most common rule so far, but the decadal averaging begins to substitute that rule. We believe that the QHS-analysis can be considered as the joint between the norms and the hazards from a bifurcation analysis viewpoint, where the causal effects of the local climate dynamics are projected into the customary timescale only at the last step. We believe that the results could be interesting to develop the fields connected with climatic change and risk assessment.

Guess-Work and Reasonings on Centennial Evolution of Surface Air Temperature in Russia. Part II: Is it Possible to Research Both Local Peculiarities and Regional Tendencies from the Bifurcation Analysis Viewpoint?

This paper is devoted to the development of the experimental bifurcation analysis in the research of local climate dynamics. In particular, we consider the dynamics of the land surface air temperature in the centennial timescale. The experimental bifurcation analysis supposes the choice of a conceptual model to demonstrate how the observable kinds of dynamical processes can be realized on the whole. We worked on the conceptual model with a variable structure (HDS-model), where the dynamics is determined by the competition between the amplitude quantization and the time quantization. The model originates from the hysteresis regulator with double synchronization (HDS-regulator) proposed in 1970’s to achieve the extreme combination of both efficiency and reliability of energy conversion processes. The HDS-model allows to consider the interplay between several periodical processes instead of chaos and quasi-periodicity in order to excuse the variety of the behaviors observed in the local climate dynamics. In particular, the intermittency seems to be the typical behavior of a local climate system from such viewpoint. Here we continue to verify the HDS-model and continue to develop the idea of the modified bifurcation diagrams to reveal the regularities within the intermittency. In particular, we first build the spatial diagram to summarize the results of the bifurcation analysis of the local climate dynamics in the centennial timescale. We assume that each effect of the regional temperature oscillations (RTO-effect) appears as a certain combination of several effects of the local temperature oscillations (LTO-effects), where each LTO-effect can be revealed by the bifurcation analysis. The possibility to build the modified bifurcation diagrams is provided by the SUC-logic aimed for the synthesis of experimental bifurcation analysis, symbolical analysis, and multidimensional data visualization under the assumption that an annual warming–cooling cycle is the unit to analyze. Since only the historical data of the temperature observations are used, then the results approach as close as possible to the real events. We believe that our research seems to be interesting to estimate the theoretically possible latent abilities and evolution of the local climate dynamics.

ADVANCED EXPERIMENTAL BIFURCATION ANALYSIS: BASELINE SCENARIO OF DYNAMICS EVOLUTION

The paper develops the advanced experimental bifurcation analysis, where the extended data acquisition, the intelligent data processing, and the modified bifurcation diagrams are combined to get useful information for different specified purposes. In particular, here we focus on how to estimate experimentally the nearest limit of the operating process stability loss without expert opinions. So, we consider the baseline scenario of dynamics evolution, where the peculiarity of the estimation is connected with the possibility to determine the uncertainty zone about the corresponding bifurcation points. The algorithms of both data acquisition and data processing are illustrated. The procedures of the algorithms are founded on the methods of the experimental researches of different nonlinear phenomena under uncertainties. The presented results can be applied to engineering design.

Experimental bifurcation analysis—Continuation for noise-contaminated zero problems

Noise contaminated zero problems involve functions that cannot be evaluated directly, but only indirectly via observations. In addition, such observations are affected by a non-deterministic observation error (noise). We investigate the application of numerical bifurcation analysis for studying the solution set of such noise contaminated zero problems, which is highly relevant in the context of equation-free analysis (coarse grained analysis) and bifurcation analysis in experiments, and develop specialized algorithms to address challenges that arise due to the presence of noise. As a working example, we demonstrate and test our algorithms on a mechanical nonlinear oscillator experiment using control based continuation, which we used as a main application and test case for development of the Coco compatible Matlab toolbox Continex that implements our algorithms.

Guess-Work and Reasonings on Centennial Evolution of Surface Air Temperature in Russia: Is it Possible to Build Bifurcation Diagrams Based on Extra-Short Local Observations?

Since the bifurcation analysis remains the basic tool to inquire into nonlinear system dynamics, then, theoretically, the bifurcation analysis should be widely used to estimate the evolution of the climate dynamics at a certain geographic point over the last centenary. But it does not occur in practice. A viewpoint how to estimate such evolution is proposed and discussed in the paper. The viewpoint can be briefly formulated as the following: the extra-short timescale, plus the novel conceptual model, plus the modified bifurcation diagrams. So, first, only the data of instrumental measurements (hereafter observations) are used to build the bifurcation diagram, correspondingly, the timescale is restricted by the maximal duration of the reliable observations. In particular, 130-years time series of the land surface air temperature in Russia are analyzed in the paper. Second, the conceptual model (HDS-model) to describe the annual warming–cooling cycle dynamics similar to the dynamics of the hysteresis regulator with double synchronization (HDS-regulator) is introduced. Correspondingly, the local climate dynamics is considered as the dynamics of the system with a variable structure. The main advantage of HDS-model is the possibility to describe quite simply the rich variety of the observable temperature evolution based on the known nonlinear phenomena by the combinations of C-bifurcations (or, in other words, border-collision bifurcations) between three conjugate periodical processes with the same period but different orders of structural changes. Then HDS-hypothesis on the climate dynamics is proposed, where the local temperature evolution is considered under the competition between both amplitude and time quantizations within the Sun–Earth–Moon system. Processing the observations follows the SUC-logic conceptions on the experimental bifurcation analysis that allows to untangle the intricate nonlinear nonstationary behavior of a local climate system by using both special sections (S) and units (U) in a certain consequence (C). As a result, the modified bifurcation diagrams to show the temperature evolution in local climate systems are built under the variation of several bifurcation parameters. The diagrams exhibit intermittency phenomena of high activity in the European part of Russia at least. Generally speaking, HDS-hypothesis gives the good approximation to answer the question whether the observable changes in the local climate dynamics are qualitative and, if yes, in what meaning?

Experimental Identification of Uncertainties in Dynamics of PWM Buck Converter

The paper is dedicated to the development of experimental bifurcation analysis and is focused on the research of uncertainties in dynamics of an operating process in application to PWM buck converters. The experimental bifurcation analysis follows the approach which allows to identify some qualitative thresholds through the successive evolution of the system behaviour caused by the complex interplay between bifurcation and noise components. In according to the approach the 2D-operating process domain is built and is divided onto three kinds of the uncertainty zones. The computational and experimental results are discussed from the practical experience concerning the ways to prevent the abnormal processes in dynamics of PWM buck converters.

Experiment on Bifurcation and Chaos in Coupled Anisochronous Self-Excited Systems: Case of Two Coupled van der Pol-Duffing Oscillators

The analog circuit implementation and the experimental bifurcation analysis of coupled anisochronous self-driven systems modelled by two mutually coupled van der Pol-Duffing oscillators are considered. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of their velocities (i.e., dissipative coupling). Interest in this problem does not decrease because of its significance and possible application in the analysis of different, biological, chemical, and electrical systems (e.g., coupled van der Pol-Duffing electrical system). Regions of quenching behavior as well as conditions for the appearance of Hopf bifurcations are analytically defined. The scenarios/routes to chaos are studied with particular emphasis on the effects of cubic nonlinearity (that is responsible for anisochronism of small oscillations). When monitoring the control parameter, various striking dynamic behaviors are found including period-doubling, symmetry-breaking, multistability, and chaos. An appropriate electronic circuit describing the coupled oscillator is designed and used for the investigations. Experimental results that are consistent with results from theoretical analyses are presented and convincingly show quenching phenomenon as well as bifurcation and chaos.

Experimental bifurcation analysis of an impact oscillator—Determining stability

We propose and investigate three different methods for assessing stability of dynamical equilibrium states during experimental bifurcation analysis, using a control-based continuation method. The idea is to modify or turn off the control at an equilibrium state and study the resulting behavior. As a proof of concept the three methods are successfully implemented and tested for a harmonically forced impact oscillator with a hardening spring nonlinearity, and controlled by electromagnetic actuators. We show that under certain conditions it is possible to quantify the instability in terms of finite-time Lyapunov exponents. As a special case we study an isolated branch in the bifurcation diagram brought into existence by a 1:3 subharmonic resonance. On this isola it is only possible to determine stability using one of the three methods, which is due to the fact that only this method guarantees that the equilibrium state can be restored after measuring stability.

Experimental bifurcation analysis of an impact oscillator—Tuning a non-invasive control scheme

We investigate a non-invasive, locally stabilizing control scheme necessary for an experimental bifurcation analysis. Our test-rig comprises a harmonically forced impact oscillator with hardening spring nonlinearity controlled by electromagnetic actuators, and serves as a prototype for electromagnetic bearings and other machinery with build-in actuators. We propose a sequence of experiments that allows one to choose optimal control-gains, filter parameters and settings for a continuation method without a priori study of a model. Depending on the algorithm for estimating the Jacobian required by Newton's method we find two almost disjoint sets of suitable control parameters. Control-based continuation succeeds reliably in producing the full bifurcation diagram including both stable and unstable equilibrium states for an appropriately tuned controller.

Study on Nonlinear Friction-Induced Vibration in Water-Lubricated Rubber Stern Tube Bearings

Nonlinear friction-induced vibration deteriorated the performance of marine rubber stern tube bearings (RSTB) in a wide variety of propulsion system. In the paper, this study was performed on the level of both practical and experimental bifurcation analysis. Hereto, an experimental set-up of SSB-100 marine stern tube bearing test machine was applied to carry on this project. The synthesis of these practical and experimental results confirmed that nonlinear friction- induced vibration was due to a subtle balance of negative damping at lower speeds or viscous friction at higher speeds. Furthermore, it was shown how these essential friction characteristics depended on physical conditions such as temperature, normal forces and the material properties in the frictional contact.