Robert Vrabel

Frequency control of singularly perturbed forced duffing's oscillator

We analyze the dynamics of the forced singularly perturbed differential equation of Duffing’s type. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative. We give some generalizations of results obtained recently by B. S. Wua, W. P. Suna, and C. W. Lim. Analytical approximations to the double-well Duffing oscillator in large amplitude oscillations (see J. Sound Vibration307 (2007), Nos. 3–5, 953–960). A new method for an analysis of the nonlinear oscillations which is based on the dynamic change of coordinates is proposed.

Active Control of Oscillation Patterns in the Presence of Multiarmed Pitchfork Structure of the Critical Manifold of Singularly Perturbed System

We analyze the possibility of control of oscillation patterns for nonlinear dynamical systems without the excitation of oscillatory inputs. We propose a general method for the partition of the space of initial states to the areas allowing active control of the stable steady-state oscillations. Furthermore, we show that the frequency of oscillations can be controlled by an appropriately positioned parameter in the mathematical model. This paper extends the knowledge of the nature of the oscillations with emphasis on its consequences for active control. The results of the analysis are numerically verified and provide the feedback for further design of oscillator circuits.

Duffing-Type Oscillator with a Bounded from above Potential in the Presence of Saddle-Center Bifurcation and Singular Perturbation: Frequency Control

We analyze the dynamics of the forced singularly perturbed differential equations of Duffing’s type with a potential that is bounded from above. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative.

Estimation of Turning Point Location of Convex Solutions for Nonlinear Second-Order Differential Equations

Abstract In this paper, we study the existence and location of turning points of the convex solutions for a certain class of the ordinary differential equations subject to the Dirichlet boundary conditions. We propose a practical and effective method for calculating the lower and upper bounds of turning point location.

Mathematical Approach to Security Risk Assessment

The goal of this paper is to provide a mathematical threat modeling methodology and a threat risk assessment tool that may assist security consultants at assessing the security risks in their protected systems/plants, nuclear power plants and stores of hazardous substances: explosive atmospheres and flammable and combustible gases and liquids, and so forth, and at building an appropriate risk mitigation policy. The probability of a penetration into the protected objects is estimated by combining the probability of the penetration by overcoming the security barriers with a vulnerability model. On the basis of the topographical placement of the protected objects, their security features, and the probability of the penetration, we propose a model of risk mitigation and effective decision making.

Local null controllability of the control-affine nonlinear systems with time-varying disturbances

The problem of local null controllability for the control-affine nonlinear systems

Stabilisation and state trajectory tracking problem for nonlinear control systems in the presence of disturbances

\dot x(t)=f(x(t))+Bu(t)+w(t),

A Heat Transfer Approach to the Calculation of Residual Power of Used Nuclear Fuel

t\in[0,T]

Three point boundary value problem for singularly perturbed semilinear differential equations.

is considered in this paper. The principal requirements on the system are that the LTI pair