Narrow Results

The initial discovery and implementation by the author of a particular kind of nonstandard finite difference (NSFD) scheme called a “Mickens finite difference” (MFD) for approximating the radial derivatives of the Laplacian in cylindrical coordinates is reviewed. The development of a similar scheme for the spherical coordinates case is also recounted. Examples of application of the schemes to several related (singular and nonsingular, linear and nonlinear) boundary value problems are given. Examples of applying Buckmire's MFD scheme to the bifurcatory, nonlinear eigenvalue problems of Bratu and Gel'fand are also presented. The results support the utility and versatility of MFD schemes for boundary value problems with singularities or bifurcations.

We study the exact multiplicity of positive solutions of the p-Laplacian Dirichlet problem.

where p > 1, φ_p (y) = |y|^p−2 y, (φ_p (u′))′ is the one-dimensional p-Laplacian, and λ > 0 is a bifurcation parameter. Assuming that satisfies (F1)-(F4), we show that the bifurcation curve has exactly one critical point, a maximum, on the (‖u‖_∞, λ)-plane. Thus we are able to determine the exact multiplicity of positive solutions. We give an interesting application for a nonlinear Dirichlet problem of polynomial nonlinearities with positive coefficients.

In this paper, the dynamic mechanism of the grazing phenomena for a dry-friction oscillator interacting with a time varying traveling surface is investigated. The theory of non-smooth dynamical systems for connectable and accessible domains is applied to this oscillator. The stick and non-stick motions are discussed through their corresponding mapping definitions. The grazing motions are presented through the initial and final switching sets, varying with external excitation parameters. The analytical prediction of grazing motion is verified through numerical simulations. This investigation provides a systematic analysis for the grazing motion in such a discontinuous dynamical system.

Turning dynamics is investigated using a 3D model that allows for simultaneous workpiece-tool deflections in response to the exertion of nonlinear regenerative force. The workpiece is modeled as a system of three rotors connected by a flexible shaft. Such a configuration enables the motion of the workpiece relative to the tool and tool motion relative to the machining surface to be three-dimensionally established as functions of spindle speed, instantaneous depth-of-cut, material removal rate and whirling. The model is explored along with its 1D counterpart, which considers only tool motions and disregards workpiece vibrations. Different stages of stability for the workpiece and the tool subject to the same cutting conditions are studied.

In the present study a single degree of freedom oscillator with clearance type non-linearity is considered. Such oscillator represents the simplest model able to analyze a single teeth gear pair, neglecting: bearings and shafts stiffness and multi mesh interactions. One of the test cases considered in the present work represents an actual gear pair that is part of a gear box of an agricultural vehicle; such gear pair gave rise to noise problems. The main gear pair characteristics (mesh stiffness and inertia) are evaluated after an accurate geometrical modelling. The meshing stiffness of the gear pair is piecewise linear and time varying (in particular periodic); it is evaluated numerically using nonlinear finite element analysis (with contact mechanics) for different positions along one mesh cycle, then it is expanded in Fourier series. A direct numerical integration approach and a smoothing technique have been considered to obtain the dynamic scenario. Bifurcation diagrams of Poincaré maps are plotted according to some sample case study from literature. Optimization procedures are proposed, in order to find optimal involute modifications that reduce gears vibration.

Two typical vibro-impact systems are considered. The periodic-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map associated with 1:4 strong resonance is obtained. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed. The results from simulation illustrate some interesting features of dynamics of vibro-impact systems. Near the bifurcation point for 1:4 strong resonance Neimark-Sacker bifurcations of periodic-impact motions and tangent and fold bifurcations of period-4 orbits are found to exist in the vibro-impact systems.

This text identifies some fractal sets generated by noninvertible maps. Two topics are presented: the fractal “box-within-a-box” bifurcations structure (described in 1975), produced by a dim 1 unimodal map, and the fractalization of basin boundaries generated by plane noninvertible maps. Some bifurcations of fractalization are described, depending on the basin boundary situation with respect to the critical curve (locus of points having two coincident preimages), and to the nature of strange repellers.

The usual assumption of isotropy of the material is shown here to be one of the reasons for the observed disparity between the theoretical and experimental values of the critical stress for plastic buckling of plates and shells. When the effect of plastic anisotropy that commonly occurs in thin-walled structures is included in the theoretical framework, the predicted load at the point of bifurcation can be significantly lower than that corresponding to the isotropic material. The present paper is intended to demonstrate the influence of plastic anisotropy on the critical stress for the elastic/plastic buckling of rectangular plates under unidirectional compression. The results are presented graphically to indicate how the critical stress is lowered by the presence of plastic anisotropy that is characterized by an R- value less than unity.

Hill-top branching is a bifurcation problem, in which the path-branching occurs exactly at a limit point of the primary path and this unusual bifurcation problem is not well realized in structural stability theory. The present study treats multiple hill-top branching. Tow or more than two bifurcation paths may be emanating from the limit point. The general solution of stiffness equations at the stability point is first described for hill-top branching. Then, the eigenvector-free pinpointing iteration to precisely locate the stability point and the branch-switching procedure to find multiple bifurcation paths will be illustrated briefly. In numerical application, a bench model for multiple hill-top branching is computed.

A computer test-bed is being established for the first-principle simulation of multi-scale structural failure under b last loads. Since t he structural failure due to explosion involves plasticity, damage, localization, thermal softening, phase transition and fragmentation, accurate constitutive models are not yet available for building materials. Also, a robust spatial discretization method is a necessity for large-scale simulation of the transition from continuous to discontinuous failure modes without invoking a fixed mesh connectivity. In this paper, the transition from continuous to discontinuous failure modes in brittle solids is identified through the bifurcation analysis of the acoustic tensor governing rate-dependent damage. A discrete constitutive model is then used to predict material failure as a decohesion or separation of continuum. To accommodate the multi-scale discontinuities involved in structural failure, the Material Point Method is developed to be a robust spatial discretization tool for the computer test-bed. As a result, the model parameters can be calibrated based on experimental data available, and routine simulation can be performed with limited computational resources. Sample problems are considered to illustrate the potential of the proposed simple procedure.

The dynamics of multiple competing political parties under spatial voting is explored. Parties are allowed to modify their positions adaptively in order to gain more votes. The parties in this model are opportunistic, in the sense that they try to maximize their share of votes regardless of any ideological position. Each party makes small corrections to its current platform in order to increase its own utility by means of the steepest ascent in the variables under its own control, i.e. by locally optimizing its own platform.

We show that in models with more than two parties bifurcations at the trivial equilibrium occur if only the voters are critical enough, that is, if they respond strongly to small changes in relative utilities. A numerical survey in a three-party model yields multiple bifurcations, multi-stability, and stable periodic attractors that arise through Hopf bifurcations. Models with more than two parties can thus differ substantially from the two-party case, where it has been shown that under the assumptions of quadratic voter utilities and complete voter participation there is always a globally stable equilibrium that coincides with the mean voter position.

In this paper, we discuss the recent existence results and compactness of solutions in the study of Nirenberg's problem in the 90's. We will discuss some new results and will outline some key ideas used in proofs of Theorems old and new. We also present some new questions.

We present some concepts within the area of dynamical systems which have been extended to non-smooth differential equations. These include the definition of Lyapunov exponents, extension of Conley-index or KAM-theory, an adaption of the Melnikov-technique for the detection of chaos and an approach to generalize Hopf bifurcation.

Bursting electrical oscillations are ubiquitous in nerve cells, and have been the focus of mathematical modeling and analysis for three decades. A key feature of these oscillations is the separation of time scales between "fast" and "slow". This separation allows one to perform a geometric singular perturbation or "fast/slow" analysis on the system. This analysis has led to many valuable insights into the dynamic mechanisms of bursting oscillations. Bursting also occurs in hormone-secreting cells of the pituitary. In most cases, however, the oscillation differs significantly from what is observed in nerve cells. In mathematical models of bursting in pituitary cells this difference is due to two factors. First, the underlying bifurcation structure of the fast subsystem is unlike that of neural bursting models. Second, the slow variable(s) is only marginally slow. As a result of these differences, some of the intuition that aids in the understanding of neural bursting can be misleading in the context of pituitary bursting. In this article I highlight some of the differences between the two classes of bursting.

This chapter is an introduction to complexity theory (encompassing chaos — a subset of complexity), a nascent domain, although, it possesses a historical root. Some fundamental properties of chaos/complexity (including complexity mindset, nonlinearity, interconnectedness, interdependency, far-from-equilibrium, butterfly effect, determinism/in-determinism, unpredictability, bifurcation, deterministic chaotic dynamic, complex dynamic, complex adaptive dynamic, dissipation, basin of attraction, attractor, chaotic attractor, strange attractor, phase space, rugged landscape, red queen race, holism, self-organization, self-transcending constructions, scale invariance, historical dependency, constructionist hypothesis and emergence), and its development are briefly examined. In particular, the similarities (sensitive dependence on initial conditions, unpredictability) and differences between deterministic chaotic systems (DCS) and complex adaptive systems (CAS) are analyzed. The edge of emergence (2^nd critical value, a new concept) is also conceived to provide a more comprehensive explanation of the complex adaptive dynamic (CAD) and emergence. Subsequently, a simplified system spectrum is introduced to illustrate the attributes, and summarize the relationships of the various categories of common systems.

Next, the recognition that human organizations are nonlinear living systems (high finite dimensionality CAS) with adaptive and thinking agents is examined. This new comprehension indicates that a re-calibration in thinking is essential. In the human world, high levels of human intelligence/consciousness (the latent impetus that is fundamentally stability-centric) drives a redefined human adaptive and evolution dynamic encompassing better potentials of self-organization or self-transcending constructions, autocatalysis, circular causation, localized spaces/networks, hysteresis, futuristic, and emergent of new order (involving a multi-layer structure and dynamic) — vividly indicating that intelligence/consciousness-centric is extremely vital. Simultaneously, complexity associated properties/characteristics in human organizations must be better scrutinized and exploited — that is, establishing appropriate complexity-intelligence linkages is a significant necessity. In this respect, nurturing of the intelligence mindset and developing the associated paradigmatic shift is inevitable.

A distinct attempt (the basic strategic approach) of the new intelligence mindset is to organize around human intrinsic intelligence — intense intelligence-intelligence linkages that exploits human intelligence/consciousness sources individually and collectively by focusing on intelligence/consciousness-centricity, complexity-centricity, network-centricity, complexity-intelligence linkages, collective intelligence, org-consciousness, complex networks, spaces of complexity (better risk management <=> new opportunities <=> higher sustainability) and prepares for punctuation points (better crisis management <=> collectively more intelligent <=> higher resilience/sustainability) concurrently — illustrating the significance of self-organizing capability and emergence-intelligence capacity. The conceptual development introduced will serve as the basic foundation of the intelligent organization (IO) theory.

The issue of symmetry and symmetry breaking is fundamental in all areas of science. Symmetry is often assimilated to order and beauty while symmetry breaking is the source of many interesting phenomena such as phase transitions, instabilities, segregation, self-organization, etc. In this contribution we review a series of sharp results of symmetry of nonnegative solutions of nonlinear elliptic differential equation associated with minimization problems on Euclidean spaces or manifolds. Nonnegative solutions of those equations are unique, a property that can also be interpreted as a rigidity result. The method relies on linear and nonlinear flows which reveal deep and robust properties of a large class of variational problems. Local results on linear instability leading to symmetry breaking and the bifurcation of non-symmetric branches of solutions are reinterpreted in a larger, global, variational picture in which our flows characterize directions of descent.

This paper explores the properties of a precessing rotor or a coupled system of precessing rotors (gyroscopes), where a special chaotic behavior in the precession angle can be found if the change of rotor angular velocity is linearly coupled by (an)holonomy to the precession angular velocity and angle. The linear coupling provides for rolling cone paths and allows spinning up and controlling the rotor simply by forcing precession at special quantum magic precession angles. The geometric phase induced by the curved path of the rotor or external curvature and part of the coupling increases with precession angle. This leads to bifurcations in coupling strength resulting in chaotic precession. As an alternative to the SO(3) matrix or quaternion representation the treatment of the three coupled rotations is here based on Euler's dynamical equations. First, the classical Magic Angle Precession (MAP) dynamics is realized by a geometric or mechanical condition (type I, transcendental solutions), where it can be experimentally demonstrated how MAP can "slave" angular degrees of freedom allowing the external control of high-frequent spin by slow oscillations. MAP is found in a commercial fitness device and is conceptually approached via Chua's electric circuit. Second, the quantum-gravitational MAP (type II, rational solutions) with discrete precession angles is analyzed on a deeper level requiring intrinsic curvature/relativistic effects adjusting holonomy to quantum numbers. Third, a macroscopic network of MAP elements is presented as a discrete-time recurrent neural network synchronizing to one common MAP I/II dynamics under special pairing and symmetry conditions (type III). In all three cases MAP can be treated as a time-discrete chaotic system with singularities given by the cosine map with several possible links to interesting applications on all scales.

Parametric roll excitations of a containership are investigated by a "Mathieu-type" roll model. The results of analytical closed-form formulas for the investigated ship are compared with the results obtained by the numerical technique of continuation. The aim of the study is twofold; firstly to extract the parametric roll boundaries of the investigated containership and secondly to validate the analytical expressions developed by international organisations.^1,2 The continuation technique implemented in our study is able to unveil in detail the dynamic portrait of the investigated dynamical system and to identify all associated bifurcations. The method of continuation in nonlinear roll dynamics not only predicts the instability roll boundaries and the steady amplitudes of roll oscillation by following the branches of stable solutions, but is also able to track the branches of unstable solutions and trace the system's bifurcations.

We present an adapted version of the classical Lyapunov-Schmidt reduction method to study, for some given integer q ≥ 1, the bifurcation of q-periodic orbits from fixed points in discrete autonomous systems. The approach puts some particular emphasis on the ℤ_q-equivariance of the reduced problem. We also discuss the relation with normal form theory, consider special cases such as equivariant, reversible or symplectic mappings, and obtain some results on the stability of the bifurcating periodic orbits. We conclude with an application of the approach to the generic bifurcation of q-periodic orbits for q ≥ 3, and showing how for q ≥ 5 Arnol'd tongues appear as an immediate consequence of the ℤ_q-equivariance.

The dynamics of laser models based on the Maxwell Bloch equation is studied. Instances of stability and chaotic behavior are investigated. Special solutions of the system one of which reduces to the Lotka Volterra system are derived. Absence of oscillating solutions in the reduced systems is studied.