Synthesis of the magnetic field using transversal 3D coil system

Magnetic field is usually generated using magnets realized as a set of simple coils. In general, those magnets generate magnetic field with nonzero components in all directions. Usually during the design process only one component of the magnetic field is taken into account, and in the optimisation procedure the currents and positions of simple coils are found to minimize the error between the axial component of the magnetic field and the required magnetic field in the ROI. In this work, it is shown that if the high quality homogeneous magnetic field is generated then indeed one may neglect non-axial components. On the other hand, if the obtained magnetic field is not homogeneous either due to design requirements of too restrictive constrains, then all other components may severely deteriorate the quality of the magnetic field. In the second part of the paper, we show how to design a 3D transversal coil system to solve problems which are intractable in the 1D case

Combination of exhaustive search and continuation method for the study of sinks in the Hénon map

Abstract-The problem of existence of stable periodic orbits (sinks) for the Hénon map in a neighborhood of classical parameter values is studied numerically. Several parameter values which sustain a sink are found. It is shown rigorously that the sinks exist. Regions of existence in the parameter space of the sinks are located using the continuation method

Automatized Search for Complex Symbolic Dynamics with Applications in the Analysis of a Simple Memristor Circuit

An automatized method to search for complex symbolic dynamics is proposed. The method can be used to show that a given dynamical system is chaotic in the topological sense. Application of this method in the analysis of a third-order memristor circuit is presented. Several examples of symbolic dynamics are constructed. Positive lower bounds for the topological entropy of an associated return map are found showing that the system is chaotic in the topological sense

Automatized Search for Complex Symbolic Dynamics with Applications in the Analysis of a Simple Memristor Circuit

An automatized method to search for complex symbolic dynamics is proposed. The method can be used to show that a given dynamical system is chaotic in the topological sense. Application of this method in the analysis of a third-order memristor circuit is presented. Several examples of symbolic dynamics are constructed. Positive lower bounds for the topological entropy of an associated return map are found showing that the system is chaotic in the topological sense. </jats:p

INTERVAL METHODS FOR RIGOROUS INVESTIGATIONS OF PERIODIC ORBITS

In this paper, we investigate the possibility of using interval arithmetic for rigorous investigations of periodic orbits in discrete-time dynamical systems with special emphasis on chaotic systems. We show that methods based on interval arithmetic when implemented properly are capable of finding all period-n cycles for considerable large n. We compare several interval methods for finding periodic orbits. We consider the interval Newton method and methods based on the Krawczyk operator and the Hansen–Sengupta operator. We also test the global versions of these three methods. We propose algorithms for computation of the invariant part and nonwandering part of a given set and for computation of the basin of attraction of stable periodic orbits, which allow reducing greatly the search space for periodic orbits. As examples we consider two-dimensional chaotic discrete-time dynamical systems, defined by the Hénon map and the Ikeda map, with the "standard" parameter values for which the chaotic behavior is observed. For both maps using the algorithms presented in this paper, we find very good approximation of the invariant part and the nonwandering part of the region enclosing the chaotic attractor observed numerically. For the Hénon map we find all cycles with period n ≤ 30 belonging to the trapping region. For the Ikeda map we find the basin of attraction of the stable fixed point and all periodic orbits with period n ≤ 15. For both systems using the number of short cycles, we estimate its topological entropy. </jats:p