Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback

In this paper, we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback

Fractional dynamical model for the generation of ECG like signals from filtered coupled Van-der Pol oscillators

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.In this paper, an incommensurate fractional order (FO) model has been proposed to generate ECG like waveforms. Earlier investigation of ECG like waveform generation is based on two identical Van-der Pol (VdP) family of oscillators, which are coupled by time delays and gains. In this paper, we suitably modify the three state equations corresponding to the nonlinear cross-product of states, time delay coupling of the two oscillators and low-pass filtering, using the concept of fractional derivatives. Our results show that a wide variety of ECG like waveforms can be simulated from the proposed generalized models, characterizing heart conditions under different physiological conditions. Such generalization of the modelling of ECG waveforms may be useful to understand the physiological process behind ECG signal generation in normal and abnormal heart conditions. Along with the proposed FO models, an optimization based approach is also presented to estimate the VdP oscillator parameters for representing a realistic ECG like signal.The work presented in this paper was supported by the E.U. ARTEMIS Joint Undertaking under the Cyclic and Person-Centric Health Management: Integrated appRoach for hOme, mobile and clinical eNvironments – (CHIRON) Project, Grant Agreement # 2009-1-100228

Application of Sparse Identification of Nonlinear Dynamics for Physics-Informed Learning

Advances in machine learning and deep neural networks has enabled complex engineering tasks like image recognition, anomaly detection, regression, and multi-objective optimization, to name but a few. The complexity of the algorithm architecture, e.g., the number of hidden layers in a deep neural network, typically grows with the complexity of the problems they are required to solve, leaving little room for interpreting (or explaining) the path that results in a specific solution. This drawback is particularly relevant for autonomous aerospace and aviation systems, where certifications require a complete understanding of the algorithm behavior in all possible scenarios. Including physics knowledge in such data-driven tools may improve the interpretability of the algorithms, thus enhancing model validation against events with low probability but relevant for system certification. Such events include, for example, spacecraft or aircraft sub-system failures, for which data may not be available in the training phase. This paper investigates a recent physics-informed learning algorithm for identification of system dynamics, and shows how the governing equations of a system can be extracted from data using sparse regression. The learned relationships can be utilized as a surrogate model which, unlike typical data-driven surrogate models, relies on the learned underlying dynamics of the system rather than large number of fitting parameters. The work shows that the algorithm can reconstruct the differential equations underlying the observed dynamics using a single trajectory when no uncertainty is involved. However, the training set size must increase when dealing with stochastic systems, e.g., nonlinear dynamics with random initial conditions

Dynamics of Oscillators Coupled by a Medium with Adaptive Impact

In this article we study the dynamics of coupled oscillators. We use mechanical metronomes that are placed over a rigid base. The base moves by a motor in a one-dimensional direction and the movements of the base follow some functions of the phases of the metronomes (in other words, it is controlled to move according to a provided function). Because of the motor and the feedback, the phases of the metronomes affect the movements of the base while on the other hand, when the base moves, it affects the phases of the metronomes in return. For a simple function for the base movement (such as $y = \gamma_{x} [r \theta_1 + (1 - r) \theta_2]$ in which $y$ is the velocity of the base, $\gamma_{x}$ is a multiplier, $r$ is a proportion and $\theta_1$ and $\theta_2$ are phases of the metronomes), we show the effects on the dynamics of the oscillators. Then we study how this function changes in time when its parameters adapt by a feedback. By numerical simulations and experimental tests, we show that the dynamic of the set of oscillators and the base tends to evolve towards a certain region. This region is close to a transition in dynamics of the oscillators; where more frequencies start to appear in the frequency spectra of the phases of the metronomes

On arbitrary-level IIR and FIR filters

A recently published method for designing IIR (infinite-impulse-response) digital filters with multilevel magnitude responses is reinterpreted from a different viewpoint. On the basis of this interpretation, techniques for extending these results to the case of finite-impulse-response (FIR) filters are developed. An advantage of the authors' method is that, when the arbitrary-level filter is implemented, its power-complementary filter, which may be required in specific applications, is obtained simultaneously. Also, by means of a tuning factor (a parameter of the scaling matrix), it is possible to generate a whole family of arbitrary-level filters

Using Periodic Systems to Model Hand-Haptic Interface Coupling

International audienceThe analysis of hand-haptic interface coupling as a whole system is an important question for the development of high quality haptic devices and their use in dynamical tasks where the haptic modality plays an important role. In this paper, we propose a periodic system, the Van der Pol equation, as a first approach for modeling hand-haptic interface coupling. In particular, we are interested in periodical gestures or tasks and the interaction models able to generate them. We analyze this system and we identify its parameters from the data position acquired during simulation. In this paper we present some preliminary results.The identification of this parameters should lead as to improve haptic systems performances