NON-LINEAR OSCILLATORS IN DISCRETE TIME: ANALYSIS AND SYNTHESIS OF DYNAMIC SYSTEMS

A physically justified method of synthesis of nonlinear oscillating systems oscillating in discrete time (DT) is proposed. Synthesized dynamic systems are used as nonlinear discrete (digital) filters and basic models of radio system elements.

Solving the problem of mathematical models overparameterization for some nonlinear oscillating systems

This study proposes a numerical-analytical method that allows us to simplify the model, which is obtained on the basis of the single observable variable of an object under the study, and which may be overparameterized. As a model, we consider a system of ordinary differential equations with polynomial right-hand sides. To solve this problem, the so-called differential model is used, that is, a system in which unknown variables are replaced by derivatives of the observed variable, and which is derived on the basis of a system under the study so that the observed variables of these systems coincide. The method of simplification of a system under the study is based on the fact that using a numerical method, a simpler differential model can be obtained. Next, an analytical transition from a simplified differential model to a simplified original system is performed. In this case, the time series error remains within given limits even for systems with deterministic chaos, despite their high sensitivity to the initial conditions.

tACS competes with ongoing oscillations for control of spike-timing in the primate brain.

Transcranial alternating current stimulation (tACS) is a promising but controversial method for modulating neural activity noninvasively. Much of the controversy revolves around the question of whether tACS can generate electric fields that are strong enough to entrain neuronal spiking activity. Here we show that what matters is not the electric field strength per se, but rather the strength of the stimulation relative to ongoing oscillatory entrainment. We recorded from single neurons in the cortex and subcortex of behaving non-human primates, while applying tACS at different frequencies and amplitudes. When neuronal activity was weakly locked to ongoing oscillations, tACS readily entrained single-neuron activity to specific stimulation phases. In contrast, neurons that were strongly locked to ongoing oscillations usually exhibited decreased entrainment during low-amplitude tACS. As this reduced entrainment is a property of many oscillating systems, attempts to impose an external rhythm on spiking activity may often yield precisely the opposite effect.

A Phase-Fitted and Amplification-Fitted Explicit Runge–Kutta–Nyström Pair for Oscillating Systems

An optimized embedded 5(3) pair of explicit Runge–Kutta–Nyström methods with four stages using phase-fitted and amplification-fitted techniques is developed in this paper. The new adapted pair can exactly integrate (except round-off errors) the common test: y″=−w2y. The local truncation error of the new method is derived, and we show that the order of convergence is maintained. The stability analysis is addressed, and we demonstrate that the developed method is absolutely stable, and thus appropriate for solving stiff problems. The numerical experiments show a better performance of the new embedded pair in comparison with other existing RKN pairs of similar characteristics.

Phototunable self-oscillating system driven by a self-winding fiber actuator

Self-oscillating systems that enable autonomous, continuous motions driven by an unchanging, constant stimulus would have significant applications in intelligent machines, advanced robotics, and biomedical devices. Despite efforts to gain self-oscillations have been made through artificial systems using responsive soft materials of gels or liquid crystal polymers, these systems are plagued with problems that restrict their practical applicability: few available oscillation modes due to limited degrees of freedom, inability to control the evolution between different modes, and failure under loading. Here we create a phototunable self-oscillating system that possesses a broad range of oscillation modes, controllable evolution between diverse modes, and loading capability. This self-oscillating system is driven by a photoactive self-winding fiber actuator designed and prepared through a twistless strategy inspired by the helix formation of plant-tendrils, which endows the system with high degrees of freedom. It enables not only controllable generation of three basic self-oscillations but also production of diverse complex oscillatory motions. Moreover, it can work continuously over 1270000 cycles without obvious fatigue, exhibiting high robustness. We envision that this system with controllable self-oscillations, loading capability, and mechanical robustness will be useful in autonomous, self-sustained machines and devices with the core feature of photo-mechanical transduction.

Dynamics of the generator with three circuits in the feedback loop. Multistability formation and transition to chaos

Background and Objectives: Studying the dynamical mechanisms of the emergence of nonlinear phenomena that are characteristic for multimode self-oscillating systems consisting of interacting oscillators and an ensemble of passive oscillators or representing active nonlinear systems with complex feedback channels is an important urgent task. The simplest example of a self-oscillating system with a complex feedback is the well-known classical van der Pol oscillator with an additional linear oscillatory circuit included in the feedback channel. We investigate the behavior of the multimode system increasing the number of oscillatory circuits in the oscillator’s feedback loop. The research in this paper can help to better understand the mechanisms of multistability formation in infinite-dimensional self-oscillating systems such as a generator with delayed feedback and a generator with distributed feedback. Materials and Methods: The system equations were derived for the electronic scheme of the self-oscillating system. To describe the existing dynamic modes by numerical simulation methods, the projections of the phase portraits and the Poincare sections were obtained. To study the mechanisms of formation of multistable states, the bifurcation analysis methods were used. Results: It was found that the mechanism underlying the multistability formation is based on a sequence of two supercritical Andronov – Hopf bifurcations and a subcritical Neymark – Saker bifurcation. Therefore, the multistability emerges as a result of gaining stability by the unstable limit set that existed before the multistability appears. Conclusion: The discovered mechanism of multistability formation opens up wide possibilities for managing the multistability, which are inaccessible for systems in which the multistability is realized through tangential bifurcations. In contrast to the tangential bifurcation, the subcritical Neymark – Sacker bifurcation assumes the existence of a limit cycle both before and after the bifurcation. Thus, it is possible to use a wide range of methods and tools to stabilize saddle limit cycles in order to control the boundaries of the multistability region in the space of control parameters of the system.

Chemical pumps and flexible sheets spontaneously form self-regulating oscillators in solution

The synchronization of self-oscillating systems is vital to various biological functions, from the coordinated contraction of heart muscle to the self-organization of slime molds. Through modeling, we design bioinspired materials systems that spontaneously form shape-changing self-oscillators, which communicate to synchronize both their temporal and spatial behavior. Here, catalytic reactions at the bottom of a fluid-filled chamber and on mobile, flexible sheets generate the energy to “pump” the surrounding fluid, which also transports the immersed sheets. The sheets exert a force on the fluid that modifies the flow, which in turn affects the shape and movement of the flexible sheets. This feedback enables a single coated (active) and even an uncoated (passive) sheet to undergo self-oscillation, displaying different oscillatory modes with increases in the catalytic reaction rate. Two sheets (active or passive) introduce excluded volume, steric interactions. This distinctive combination of the hydrodynamic, fluid–structure, and steric interactions causes the sheets to form coupled oscillators, whose motion is synchronized in time and space. We develop a heuristic model that rationalizes this behavior. These coupled self-oscillators exhibit rich and tunable phase dynamics, which depends on the sheets’ initial placement, coverage by catalyst and relative size. Moreover, through variations in the reactant concentration, the system can switch between the different oscillatory modes. This breadth of dynamic behavior expands the functionality of the coupled oscillators, enabling soft robots to display a variety of self-sustained, self-regulating moves.