Modeling processes that provide an avalanche of innovation growth

This paper proposes an addition to Kondratyev’s theory of the emergence of innovations in long cycles. Regularities of the emergence of crisis phenomena and the concept of “avalanche-like growth of innovations” are considered. The study investigated the innovation peaks occurring in the middle of the depression phase, followed by the growth stages of economic activity after a certain period of time. Research has shown that the active emergence of innovations, which we have called the “snowballing growth of innovations,” falls in the middle of the depression phase. The authors investigated and supplemented the theory of the triggering effect of depression, which is similar to the action of the trigger, which results in an “avalanche-like growth of innovations”. To describe the processes associated with resonance and trigger effects, the authors propose to use the parametric resonance model and the Mathieu equation. With the help of mathematical modeling of innovation processes, a more accurate description of the periodic change in the number of innovations over time is possible, namely, the “avalanche-like growth of innovations”.

Parametric Frequency Analysis of Mathieu–Duffing Equation

The classic linear Mathieu equation is one of the archetypical differential equations which has been studied frequently by employing different analytical and numerical methods. The Mathieu equation with cubic nonlinear term, also known as Mathieu–Duffing equation, is one of the many extensions of the classic Mathieu equation. Nonlinear characteristics of such equation have been investigated in many papers. Specifically, the method of multiple scale has been used to demonstrate the pitchfork bifurcation associated with stability change around the first unstable tongue and Lie transform has been used to demonstrate the subharmonic bifurcation for relatively small values of the undamped natural frequency. In these works, the resulting bifurcation diagram is represented in the parameter space of the undamped natural frequency where a constant value is allocated to the parametric frequency. Alternatively, this paper demonstrates how the Poincaré–Lindstedt method can be used to formulate pitchfork bifurcation around the first unstable tongue. Further, it is shown how higher order terms can be included in the perturbation analysis to formulate pitchfork bifurcation around the second tongue, and also subharmonic bifurcations for relatively high values of parametric frequency. This approach enables us to demonstrate the resulting global bifurcation diagram in the parameter space of parametric frequency, which is beneficial in the bifurcation analysis of systems with constant undamped natural frequency, when the frequency of the parametric force can vary. Finally, the analytical approximations are verified by employing the numerical integration along with Poincaré map and phase portraits.

Group interaction of robots with an arbitrary kinematic diagram of manipulators during synchronous execution of technological operations

The stabilization of the manipulated object near the position of static equilibrium under kinematic and force disturbances during cooperative transportation by two robots of arbitrary structure is considered. The dynamic balance of an object in grippers is described by the Mathieu equation. The controlled executive system of the coupled robot provides adjustment of the manipulation system with minimal potential energy in the grippers. A variant of the optimal control in terms of speed is considered when transferring the system to a position close to static equilibrium. Keywords: robot, group control, controlled generalized coordinates, continuity of connections, cooperative work. [email protected]

A Simple Approach for the Computation of Lyapunov-Floquet Transformations for the Mathieu Equation

Lyapunov-Floquet (L-F) transformations reduce linear ordinary differential equations with time-periodic coefficients (so-called linear time-periodic systems) to equations with constant coefficients. The present work proposes a simple approach to construct L-F transformations. The solution of a linear time-periodic system can be expressed as a product of an exponential term and a periodic term. Using this Floquet form of a solution, the ordinary differential equation corresponding to a linear time-periodic system reduces to an eigenvalue problem. Next, eigenanalysis is performed to obtain the general solution and subsequently, the state transition matrix of the time-periodic system is constructed. Then, the Lyapunov-Floquet theorem is used to compute L-F transformation. The inverse of L-F transformation is determined by defining the adjoint system to the time-periodic system. Mathieu equation is investigated in this work and L-F transformations and their inverse are generated for stable and unstable cases. These transformations are very useful in the design of controllers using time-invariant methods and in the bifurcation studies of nonlinear time-periodic systems.

OPTIMIZATION OF ULTRASOUND MEDICAL PARAMETERS TOOLS

Для многих видов медицинских вмешательств требуется применение ультразвуковых инструментов с различными характеристиками. Используются инструменты, совершающие продольные колебания, значительно реже - инструменты с изгибами и крутильными колебаниями, либо достаточно длинные ультразвуковые медицинские инструменты, либо короткие, но тонкие. В таких инструментах часто наблюдается так называемая динамическая потеря устойчивости, когда прямолинейный инструмент, совершающий продольные колебания, внезапно начинает совершать изгибные колебания, амплитуда которых бывает настолько высока, что приводит к разрушению инструмента. Такое явление также называют параметрическим резонансом ультразвуковых инструментов. Цель статьи - анализ условий и параметров, позволяющих минимизировать травматичность применения ультразвуковых медицинских инструментов, исследование в динамике устойчивости ультразвуковых низкочастотных медицинских инструментов. Для определения оптимального набора параметров динамической устойчивости изгибных колебаний ультразвуковых низкочастотных медицинских инструментов используется уравнение Матье-Хилла. В этом аспекте решение задачи сводится к определению: 1) границ областей неустойчивости уравнения Матье; 2) границ областей неустойчивости при разных значениях коэффициента возбуждения; 3) границ областей неустойчивости с применением метода малого параметра. Для исследования динамической устойчивости уравнения колебаний прямолинейного стержня переменного сечения достаточно выполнить расчет коэффициентов уравнения Матье и использовать диаграмму Айнса-Стретта для нахождения точек попадания в область устойчивости. Результаты расчетов показали, что инструменты, изготовленные из титана, обладают высокой динамической устойчивостью, что практически исключает вероятность их разрушения при проведении медицинских операций. Полученные характеристики медицинских инструментов указывают на эффективность их применения в медицинской практике Many types of medical interventions require the use of ultrasound instruments with different characteristics. Instruments that perform longitudinal vibrations are used, much less often-instruments with bends and torsional vibrations, or rather long ultrasound medical instruments, or short, but thin. In such instruments, the so-called dynamic loss of stability is often observed, when a straight-line tool that performs longitudinal vibrations suddenly begins to make bending vibrations, the amplitude of which is so high that it leads to the destruction of the tool. This phenomenon is also called parametric resonance of ultrasonic instruments. The purpose of the article is to analyze the conditions and parameters that allow minimizing the traumaticity of the use of ultrasonic medical instruments, to study the dynamics of the stability of ultrasonic low-frequency medical instruments. The Mathieu-Hill equation is used to determine the optimal set of parameters for the dynamic stability of bending vibrations of ultrasonic low-frequency medical instruments. In this aspect, the solution of the problem is reduced to the definition of: 1) the boundaries of the instability regions of the Mathieu equation; 2) the boundaries of the instability regions at different values of the excitation coefficient; 3) the boundaries of the instability regions using the small parameter method. To study the dynamic stability of the equation of oscillations of a rectilinear rod of variable cross-section, it is sufficient to calculate the coefficients of the Mathieu equation and use the Ains-Strett diagram to find the points of falling into the stability region. The results of the calculations showed that the instruments made of titanium have a high dynamic stability, which practically eliminates the possibility of their destruction during medical operations. The obtained characteristics of medical instruments indicate the effectiveness of their use in medical practice

A Singularly P-Stable Multi-Derivative Predictor Method for the Numerical Solution of Second-Order Ordinary Differential Equations

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.