Wada index based on the weighted and truncated Shannon entropy

Nonlinear Dynamics - The Wada index based on the weighted and truncated Shannon entropy is presented in this paper. The proposed Wada index can detect if a given basin boundary is a Wada boundary....     

The effect of dynamical self-orientation and its applicability for identification of natural frequencies

The effect of dynamical self-orientation and its applicability for the identification of natural frequencies of the investigated systems is demonstrated in this paper. Unidirectional vibration exciter is fixed to the investigated systems via a pivot link and can rotate around it. It is shown that the exciter changes its orientation in the steady state motion mode when the frequency of excitation sweeps over the fundamental frequency of the examined system. Approximate analytical analysis of the discrete system illustrates the basic principle of the effect of dynamical self-orientation. Numerical analysis of both the discrete and different continuous elastic systems confirms the applicability of the effect of self-orientation for the identification of natural frequencies.     

Fractal dimension and Wada measure revisited: no straightforward relationships in NDDS

An extended Newton’s discrete dynamical system with a complex control parameter is investigated in this paper. A novel computational algorithm is introduced for the evaluation of Wada measure. A nontrivial relationship between the fractal dimension and the Wada measure is revealed in NDDS. It is demonstrated that the reduction of the fractal dimension of basin boundaries of coexisting attractors does not automatically imply a lower Wada measure of these boundaries. Computational experiments are used to illustrate what impact the complexity of the relationship between fractal dimension and Wada measure does have in practical applications.     

Algebraic approach for the exploration of the onset of chaos in discrete nonlinear dynamical systems

An algebraic approach based on the rank of a sequence is proposed for the exploration of the onset of chaos in discrete nonlinear dynamical systems. The rank of the partial solution is identified and a special technique based on Hankel matrices is used to decompose the solution into algebraic primitives comprising roots of the modified characteristic equation. The distribution of roots describes the dynamical complexity of a solution and is used to explore properties of the nonlinear system and the onset of chaos.     

Hybrid numerical—experimental approach for investigation of dynamics of microcantilever relay system

The time average holography measurements of the vibrating microelectromechanical switch (MEMS) were performed in this study. Experimental measurement results exhibit good agreement with computer generated holographic interferogram analysis. The validation of experimental investigations versus numerical analysis provides the necessary background to analyze the dynamical characteristics of micromechanical systems in virtual numerical environments. Direct application of fringe counting techniques for reconstruction of motion from time average holograms cannot be straightforward if the analyzed micromechanical systems contain motion limiters. Modifications of a classical time average holographic technique enable qualitative analysis of MEMS and can be applied for investigation of dynamical properties of much broader classes of MEMS systems.     

Contrast enhancement of time-averaged fringes based on moving average mapping functions

Moving average contrast enhancement techniques are applied for visualization of time-averaged fringes produced by time average projection moiré. The complexity of the problem is based on the fact that grayscale levels at centerlines of fringes depend from the geometrical location of these fringes. Moreover, moiré grating geometry determines the direction of sensitivity to dynamic deflections. Standard fringe visualization methods fail to produce interpretable results. The developed pixel-based analysis techniques enable efficient reconstruction of projected fringes.     

Plotting isoclinics for hybrid photoelasticity and finite element analysis

Displacement-based finite element method formulations are coupled with stress-based photoelasticity analysis. As the stress field is discontinuous at the interelement boundaries, the introduced smoothing procedure enables the generation of high-quality digital images acceptable for hybird experimental-numerical techniques. The proposed methods are applicable for the analysis of static and dynamic results of experimental photoelasticity.     

The explosive divergence in iterative maps of matrices

The effect of explosive divergence in generalized iterative maps of matrices is defined and described using formal algebraic techniques. It is shown that the effect of explosive divergence can be observed in an iterative map of square matrices of order 2 if and only if the matrix of initial conditions is a nilpotent matrix and the Lyapunov exponent of the corresponding scalar iterative map is greater than zero. Computational experiments with the logistic map and the circle map are used to illustrate the effect of explosive divergence occurring in iterative maps of matrices.     

A pseudo-stable structure in a completely invertible bouncer system

It is shown that a pseudo-stable structure of non-asymptotic convergence may exist in a completely invertible bouncing ball model. Visualization of the pattern of H-ranks helps to identify this structure. It appears that this structure is similar to the stable manifold of non-invertible nonlinear maps which govern the non-asymptotic convergence to unstable periodic orbits. But this convergence to the unstable repeller of the bouncing ball problem is only temporary since non-asymptotic convergence cannot exist in completely invertible maps. This nonlinear effect is exploited for temporary stabilization of unstable periodic orbits in completely reversible nonlinear maps.