СКРЫТЫЕ АТТРАКТОРЫ НЕКОТОРЫХ МУЛЬТИСТАБИЛЬНЫХ СИСТЕМ С БЕСКОНЕЧНЫМ ЧИСЛОМ СОСТОЯНИЙ РАВНОВЕСИЯ

It is well known that mathematically simple systems of nonlinear differential equations can exhibit chaotic behavior. Detection of attractors of chaotic systems is an important problem of nonlinear dynamics. Results of recent researches have made it possible to introduce the following classification of periodic and chaotic attractors depending on the presence of neighborhood of equilibrium into their basin of attraction – self-excited and hidden attractors. The presence of hidden attractors in dynamical systems has received considerable attention to both theoretical and applied research of this phenomenon. Revealing of hidden attractors in real engineering systems is extremely important, because it allows predicting the unexpected and potentially dangerous system response to perturbations in its structure. In the past three years after discovering by S. Jafari and J. C. Sprott chaotic system with a line and a plane of equilibrium with hidden attractors there has been much attention to systems with uncountable or infinite equilibria. In this paper it is offered new models of control systems with an infinite number of equilibrium possessing hidden chaotic attractors: a piecewise-linear system with a locally stable segment of equilibrium and a system with periodic nonlinearity and infinite number of equilibrium points. The original analytical-numerical method developed by the author is applied to search hidden attractors in investigated systems.Хорошо известно, что математически простые нелинейные системы дифференциальных уравнений могут демонстрировать хаотическое поведение. Обнаружение аттракторов хаотических систем – важная проблема нелинейной динамики. Результаты недавних исследований позволили ввести следующую классификацию периодических и хаотических аттракторов в зависимости от наличия окрестностей состояний равновесия в их области притяжения – самовозбуждающиеся и скрытые аттракторы. Присутствие скрытых аттракторов в динамических системах привлекло пристальное внимание, как к теоретическим, так и к прикладным исследованиям этого феномена. Выявление скрытых аттракторов в реальных инженерных системах чрезвычайно важно, поскольку оно позволяет предсказать неожиданные и потенциально опасные ответы системы на возмущения ее структуры. В последние три года, после обнаружения S. Jafari и J. C. Sprott хаотических систем с линией и плоскостью состояний равновесия, имеющих скрытые аттракторы, возрос интерес к системам, обладающим несчетным или бесконечным числом состояний равновесия. В настоящей работе предложены новые модели систем управления с бесконечным числом состояний равновесия, обладающие скрытыми аттракторами: кусочно-линейная система с локально устойчивым отрезком покоя и система с периодической нелинейностью и бесконечным числом состояний равновесия. Для поиска скрытых аттракторов исследуемых систем применен предложенный автором оригинальный аналитико-численный метод

Experimental and theoretical research of the interaction between high-strength supercavitation impactors and monolithic barriers in water

The article describes experimental and theoretical research of the interaction between supercavitating impactors and underwater aluminum alloy and steel barriers. Strong alloys are used for making impactors. An experimental research technique based on a high-velocity hydro-ballistic complex was developed. Mathematical simulation of the collision the impactor and barrier is based on the continuum mechanics inclusive of the deformation and destruction of interacting bodies. Calculated and experimental data on the ultimate penetration thickness of barriers made of aluminum alloy D16T and steel for the developed supercavitating impactor are obtained

High-speed impact of the metal projectile on the barrier containing porous corundum-based ceramics with chemically active filler

The paper presents a calculation-experimental study on high-speed interaction of the metal projectile with a combined barrier made of porous corundum-based ceramics filled with chemically active composition (sulfur, nitrate of potash) in the wide range of speeds. A mathematical behavior model of porous corundum-based ceramics with chemically active filler is developed within the scope of mechanics of continuous media taking into account the energy embedding from a possible chemical reaction between a projectile metal and filler at high-speed impact. Essential embedding of inlet heat is not observed in the considered range of impact speeds (2.5 … 8 km/s)

Theory of differential inclusions and its application in mechanics

The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torque characteristic is asymmetrical. Problem of sudden load change is studied. Analytical methods of investigation of systems with such asymmetrical friction based on the use of Lyapunov functions are demonstrated. The Watt governor and Chua system are considered to show different aspects of computer modeling of discontinuous systems

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

β1-integrins signaling and mammary tumor progression in transgenic mouse models: implications for human breast cancer

Consistent with their essential role in cell adhesion to the extracellular matrix, integrins and their associated signaling pathways have been shown to be involved in cell proliferation, migration, invasion and survival, processes required in both tumorigenesis and metastasis. β1-integrins represent the predominantly expressed integrins in mammary epithelial cells and have been proven crucial for mammary gland development and differentiation. Here we provide an overview of the studies that have used transgenic mouse models of mammary tumorigenesis to establish β1-integrin as a critical mediator of breast cancer progression and thereby as a potential therapeutic target for the development of new anticancer strategies