9704554 Levi The first part of the project part deals with the study of mathematical aspects of Josephson junctions. It was discovered recently that the junctions occur naturally in certain crystals where superconducting layers interlace with insulating layers. It is proposed to derive and to study a mathematical model of such crystals and to develop theory of traveling waves in discrete media. In the second part of the project it is proposed to produce the qualitative analysis of systems of relaxation oscillators of a recently discovered type. The third part of the project deals with various applications of the investigator's recent observation on the role of curvature in high- frequency averaging. These applications include the study of the motion of rigid bodies (such as particles embedded in fluid) or coupled systems of particles, such as molecules subjected to rapidly varying force fields. Another application includes the behavior of a resonant medium subjected to an electromagnetic pulse. Finally, the last part of the project deals with establishing a connection between two classes of systems: the geodesic motion on a surface on the one hand and the billiard motion inside that surface on the other. The first of the four parts of the project deals with the study of Josephson junctions -- superconducting devices whose properties offer great potential for applications. The advantages of Josephson junctions lie in their small size and in low energy consumption, which allow for a great reduction of the circuit size, but the disadvantage is the low power output. To overcome that disadvantage, the junctions can be put in arrays. Remarkably, such arrays occur naturally: some crystals were discovered to consist of stacked Josephson junction layers. This recent discovery offers potential new applications, notably for use in interfaces between ultrafast optical communication lines and electric circuits as well as for generation of high frequency oscillations for use in communications and radio-astronomical observations. Understanding the behavior of such arrays takes on a crucial importance. The project proposed here aims at such understanding. The second part of the project addresses relaxation oscillators - these occur in biological and electronic systems whose oscillations consist of slow drifts punctuated by fast jumps. A large class of such oscillators exhibiting chaotic behavior was discovered recently, and an isolated oscillator analyzed. Understanding systems of coupled oscillators (the goal of this project) offers great challenges and is of importance in some biological and electric systems. In the third part of this effort, the dynamical effects of high-frequency oscillations will be explored. Such effects underlie the operation of the Paul trap (the inventor was awarded a Nobel prize) and the more recent "laser tweezers" which allow to move objects inside the cell without damaging it. This research will extend mathematical analysis of these phenomena with the aim of possible applications which include vibrational control as well as environmental applications such as separation processes (removing particles from the air, etc.) In the fourth part of the project it is proposed to establish a connection between two century-old geometry problems which were developed in parallel but without a direct connection, by showing that one is a limiting case of another. Both of these problems played a central role in the development of the theory of dynamical systems, and thus indirectly but crucially in the development of particle accelerators, in astronomy and in optics.
