Synthesizing attractors of Hindmarsh-Rose neuronal systems

In this paper a periodic parameter switching scheme is applied to the Hindmarsh-Rose neuronal system to synthesize certain attractors. Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh-Rose neuronal system and matches the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This feature allows us to imagine that living beings are able to maintain vital behavior while the control parameter switches so that their dynamical behavior is suitable for the given environment.Comment: published in Nonlinear Dynamic

Approach of a class of discontinuous dynamical systems of fractional order: existence of the solutions

In this letter we are concerned with the possibility to approach the existence of solutions to a class of discontinuous dynamical systems of fractional order. In this purpose, the underlying initial value problem is transformed into a fractional set-valued problem. Next, the Cellina's Theorem is applied leading to a single-valued continuous initial value problem of fractional order. The existence of solutions is assured by a P\'{e}ano like theorem for ordinary differential equations of fractional order.Comment: accepted IJBC, 5 pages, 1 figur

A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set

The external rays of the Mandelbrot set are a valuable graphic tool in order to study this set. They are drawn using computer programs starting from the Böttcher coordinate. However, the drawing of an external ray cannot be completed because it reaches a point from which the drawing tool cannot continue drawing. This point is influenced by the resolution of the standard for floating-point computation used by the drawing program. The IEEE 754 Standard for Floating-Point Arithmetic is the most widely used standard for floating-point computation, and we analyze the possibilities of the quadruple 128 bits format of the current IEEE 754-2008 Standard in order to draw external rays. When the drawing is not possible, due to a lack of resolution of this standard, we introduce a method to draw external rays based on the escape lines and Bézier curves

Bifurcation Boundary Conditions for Switching DC-DC Converters Under Constant On-Time Control

Sampled-data analysis and harmonic balance analysis are applied to analyze switching DC-DC converters under constant on-time control. Design-oriented boundary conditions for the period-doubling bifurcation and the saddle-node bifurcation are derived. The required ramp slope to avoid the bifurcations and the assigned pole locations associated with the ramp are also derived. The derived boundary conditions are more general and accurate than those recently obtained. Those recently obtained boundary conditions become special cases under the general modeling approach presented in this paper. Different analyses give different perspectives on the system dynamics and complement each other. Under the sampled-data analysis, the boundary conditions are expressed in terms of signal slopes and the ramp slope. Under the harmonic balance analysis, the boundary conditions are expressed in terms of signal harmonics. The derived boundary conditions are useful for a designer to design a converter to avoid the occurrence of the period-doubling bifurcation and the saddle-node bifurcation.Comment: Submitted to International Journal of Circuit Theory and Applications on August 10, 2011; Manuscript ID: CTA-11-016

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

Suppressing chaos in discontinuous systems of fractional order by active control

In this paper, a chaos control algorithm for a class of piece-wise continuous chaotic systems of fractional order, in the Caputo sense, is proposed. With the aid of Filippov’s convex regularization and via differential inclusions, the underlying discontinuous initial value problem is first recast in terms of a set-valued problem and hence it is continuously approximated by using Cellina’s Theorem for differential inclusions. For chaos control, an active control technique is implemented so that the unstable equilibria become stable. As example, Shimizu–Morioka’s system is considered. Numerical simulations are obtained by means of the Adams–Bashforth–Moulton method for differential equations of fractional-order