Vol. 2, No. 2 (2013): Regular Issue

The aim of this work is to study the dynamics of pendulum driven through its pivot moving in both horizontal and vertical directions. It expands the results obtained for the parametric pendulum by Lenci et al. to two other cases, i.e. the elliptically excited pendulum and the pendulum, with an inclined rectilinear base motion (the tilted pendulum). Here we derive approximate analytical expressions representing the position of the saddle-node bifurcation associated with period-1 rotations in the excitation amplitude/frequency plane in the presence of damping by using the perturbation method proposed by Lenci et al. This includes development of a procedure for deducing expressions for the period doubling, creating a pair of stable period-2 rotational attractors. The obtained approximations are plotted on the excitation parameters plane and compared with numerical results. Simple Padé approximations for the analytical expressions relating to the position of the saddle-node bifurcation are also obtained.

In this paper we examine the chaotic regimes of a variety of recently discovered hyperchaotic systems using a completely novel geometrical interpretation of the Competitive Modes analysis as simple criteria to map out the spatial location and extent, as well as the rough general shape, of the system attractor for any parameter sets corresponding to chaos. The accuracy of this mapping adds further evidence to the growing body of recent work on the correctness and usefulness of these Competitive Modes conjectures. Indeed, this may be taken as an ’a posteriori’ validation of the Competitive Modes conjectures.

Many random signals with clearly expressed trends can have selfsimilar properties. In order to see this self-similar property new presentation of signals is suggested. A novel algorithm for inverse solution of the scaling equation is developed. This original algorithm allows finding the scaling parameters, the corresponding power-law exponent and the unknown log-periodic function from the fitting procedure. The effectiveness of algorithm is tested in financial data revealing season fluctuations of annual, monthly and weekly prices. The general recommendations are given that allow the verification of this algorithm in general data series.

We study a multi point boundary value problem for a fractional order differential inclusion involving a nonconvex set-valued map. We establish a Filippov type existence theorem and we prove the arcwise connectedness of the solution set of the problem considered.

In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.

An analytical method is presented to determine the effect of adding different materials at one of the edges of an annular rotating disk on its in-plane natural frequencies and critical speeds. The proposed analysis is based on the linear in-plane free vibration of a compound disk with material discontinuity, by adopting the two-dimensional plane stress theory. The frequency equation was achieved by satisfying the compatibilities of the displacements and stresses at the interfaces of the different segments. The materials used in each segments of the disk are assumed to be homogenous,elastic, and isotropic. Furthermore, the annular disk is considered to be clamped at the inner side and free at the outer edge with a radius ratio of 0.3, and rotates with a constant angular speed. The variation of non-dimensional natural frequencies in fixed coordinates for different modes and different segment radiuses at the inner or outer side with respect to speed of rotation are computed. Presented results indicated that by adding additional segment, undesirable natural frequencies of the rotating disk can be modified to be within the acceptable range.

Models of evolutionary dynamics are often approached via the replicator equation, which in its standard form is given by ˙ xi = xi ( fi (x)−φ ) , i = 1, . . . ,n, where xi is the frequency of strategy i, fi is its fitness, and φ = Σn i=1 xi fi is the average fitness. A game-theoretic aspect is introduced to the model via the payoff matrix A by taking fi(x) = (A · x)i. This model is based on the exponential model of population growth, ˙ xi = xi fi, with φ introduced in order both to hold the total population constant and to model competition between strategies. We analyze the dynamics of analogous models for the replicator equation of the form ˙ xi = g(xi)( fi −φ ), for selected growth functions g.