Sliding and transversal motions on an inclined boundary in a periodically forced discontinuous system

In this paper, sliding and transversal motions on the boundary in the periodically driven, discontinuous dynamical system is investigated. The simple inclined straight line boundary in phase space is considered as a control law for such a dynamical system to switch. The normal vector field for a flow switching on the separation boundary is adopted to develop the analytical conditions, and the corresponding transversality conditions of a flow to the boundary are obtained. The conditions of sliding and grazing flows to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the corresponding local stability and bifurcation analysis of the periodic motion are carried out. Numerical illustrations of periodic motions with and without sliding on the boundary are given. The local stability analysis cannot provide the proper prediction of the sliding and grazing motions in discontinuous dynamical systems. Therefore, the normal vector fields of periodic flows are presented, and the normal vector fields on the switching boundary points give the analytical criteria for sliding and transversality of motions.     

On flow switching bifurcations in discontinuous dynamical systems

In paper, the sliding dynamics on the separation boundary is discussed based on the set-valued vector field theory. From vector fields in the neighborhood of a specific separation boundary, the passability of the flow from the one domain into another one is further discussed. The switching bifurcation conditions from the passable boundary to the non-passable boundary are developed. The sliding flow fragmentation on the separation boundary surface is also presented. The normal vector product field function is introduced to determine the switching bifurcation and sliding fragmentation.     

Flow switchability and periodic motions in a periodically forced,discontinuous dynamical system

This paper presents the switchability of a flow from one domain into another one in the periodically forced, discontinuous dynamical system. The inclined line boundary in phase space is used for the dynamical system to switch. The normal vector field product for flow switching on the separation boundary is introduced. The passability condition of a flow to the separation boundary is achieved through such a normal vector field product, and the sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions in such a discontinuous system are predicted analytically, and the corresponding local stability and bifurcation analysis are carried out. With the analytical conditions of grazing and sliding motions, the parameter maps of specific motions are developed. Illustrations of periodic and chaotic motions are given, and the normal vector fields are presented to show the analytical criteria. This investigation may help one better understand the sliding mode control. The methodology presented in this paper can be applied to discontinuous, nonlinear systems.     

Long Eddies in Sheared Flows

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity ω(ψ) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (1) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form ωαyⁿ, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar.     

Asymptotic and numerical solutions of three‐dimensional boundary‐layer flow past a moving wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows.     

New approach in dynamics of regenerative chatter research of turning

In this paper, regenerative chatter phenomena in a turning process is discussed from impulsive dynamical point of view. By introducing the instantaneous pulse when vibration occurs and the vibratory condition set, we optimize the models and present a certain kind of second-order impulsive differential systems, which is a specific discontinuous dynamical system. Then we search for the general results of the nonoccurrence of chatter phenomena by discussing the number of the vibration pulse times, utilizing the method of flow theory in discontinuous systems and transversal property at the boundary. Our results give a convenient way to estimate the available parameters to keep the turning process stable.     

Numerical and asymptotic study of non‐axisymmetric magnetohydrodynamic boundary layer stagnation‐point flows

Both numerical and asymptotic analyses are performed to study the similarity solutions of three‐dimensional boundary‐layer viscous stagnation point flow in the presence of a uniform magnetic field. The three‐dimensional boundary‐layer is analyzed in a non‐axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller‐box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far‐field behavior and in the limit of large shear‐to‐strain‐rate parameter (λ). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and λ. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near‐field (due to viscous forces) and far‐field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary‐layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

Control of the Planar Takens–Bogdanov Bifurcation with Applications

It is well-known that on a versal deformation of the Takens–Bogdanov bifurcation is possible to find dynamical systems that undergo saddle-node, Hopf, and homoclinic bifurcations. In this document a nonlinear control system in the plane is considered, whose nominal vector field has a double-zero eigenvalue, and then the idea is to find under which conditions there exists a scalar control law such that be possible establish a priori, that the closed-loop system undergoes any of the three bifurcations: saddle-node, Hopf or homoclinic. We will say then that such system undergoes the controllable Takens–Bogdanov bifurcation. Applications of this result to the averaged forced van der Pol oscillator, a population dynamics, and adaptive control systems are discussed.     

On zero-sum -flows of graphs

For a graph G, a zero-sum flow is an assignment of non-zero real numbers on the edges of G such that the total sum of all edges incident with any vertex of G is zero. A zero-sum k-flow for a graph G is a zero-sum flow with labels from the set {±1,…,±(k-1)}. In this paper for a graph G, a necessary and sufficient condition for the existence of zero-sum flow is given. We conjecture that if G is a graph with a zero-sum flow, then G has a zero-sum 6-flow. It is shown that the conjecture is true for 2-edge connected bipartite graphs, and every r-regular graph with r even, r>2, or r=3.     

Kernel density in the study of the strong stability of the queueing system

We use kernel density with correction of boundary effects to study the strong stability of the M/M/1 system after perturbation of arrival flow (respectively service times), to evaluate the proximity of G/M/1 (respectively M/G/1) and M/M/1 systems when the distribution G is unknown. Simulation studies are performed to support the results.     

Compact abelian group extensions of discrete dynamical systems

Summary This paper introduces the notion of a free G extension of a dynamical system where G is a compact abelian group. The concept is closely allied to that of generalised discrete spectrum (which includes Abramov's quasi-discrete spectrum as a special case). We give necessary and sufficient conditions for a G extension of a minimal (uniquely ergodic) dynamical system to be minimal (uniquely ergodic) and show that in a certain sense a general G extension lifts these properties. Stable G-extensions always lift these properties if the underlying space is connected. This fact is then used to characterise all uniquely ergodic and minimal affine transformations of a certain three dimensional nilmanifold. The rest of the paper is devoted to the exhibition of group invariants for systems with generalised discrete spectrum. In particular it is shown that such systems always have a compact abelian group as underlying space. A lemma which facilitates this result gives necessary and sufficient conditions for a connected G-extension of a compact abelian group to be a compact abelian group.     

Isolating blocks for Morse flows

We present a constructive general procedure to build Morse flows on n-dimensional isolating blocks respecting given dynamical and homological boundary data recorded in abstract Lyapunov semi-graphs. Moreover, we prove a decomposition theorem for handles which, together with a special class of gluings, insures that this construction not only preserves the given ranks of the homology Conley indices, but it is also optimal in the sense that no other Morse flow can preserve this index with fewer singularities.      

A periodically forced,piecewise linear system. Part I: Local singularity and grazing bifurcation

A methodology for the local singularity of non-smooth dynamical systems is systematically presented in this paper, and a periodically forced, piecewise linear system is investigated as a sample problem to demonstrate the methodology. The sliding dynamics along the separation boundary are investigated through the differential inclusion theory. For this sample problem, a perturbation method is introduced to determine the singularity of the sliding dynamics on the separation boundary. The criteria for grazing bifurcation are presented mathematically and numerically. The grazing flows are illustrated numerically. This methodology can be very easily applied to predict grazing motions in other non-smooth dynamical systems. The fragmentation of the strange attractors of chaotic motion will be presented in the second part of this work.     

Zwei Klassen vollständiger Funktionensysteme zur Behandlung der Randwertaufgaben der Schwingungsgleichung ΔU + k2U = 0

In this paper we present new methods to solve the classical Dirichlet and Neumann problems for ΔU + k²U = 0. We prove that the solutions of this equation for a region S containing G restricted to G are dense in L²(?G). Introducing a basis in the space of solutions for S we find a complete orthogonal system in L²(?G) which can be used to solve the boundary value problems by means of approximation in the Hilbertspace norm. Regularity estimates lead to series expansions in G. The well-known basis systems obtained by separation of variables thus may be used for every regular region without the very special geometric restrictions. Another class of basis systems may be obtained in analogy to the Runge. theorems by considering types of singularity functions.     

Nonsmooth one-dimensional maps: some basic concepts and definitions

The main purpose of the present survey is to contribute to the theory of dynamical systems defined by one-dimensional piecewise monotone maps. We recall some definitions known from the theory of smooth maps, which are applicable to piecewise smooth ones, and discuss the notions specific for the considered class of maps. To keep the presentation clear for the researchers working in other fields, especially in applications, many examples are provided. We focus mainly on the notions and concepts which are used for the investigation of various kinds of attractors of a map and related bifurcation structures observed in its parameter space.     

Periodic motions in a simplified brake system with a periodic excitation

In this paper, periodic motions for a simplified brake system under a periodical excitation are investigated, and the motion switchability on the discontinuous boundary is discussed through the theory of discontinuous dynamical systems. The onset and vanishing of periodic motions are discussed through the bifurcation and grazing analyses. Based on the discontinuous boundary, the switching planes and the basic mappings are introduced, and the mapping structures for periodic motions are developed. From the mapping structures, the periodic motions are analytically predicted and the corresponding local stability and bifurcation analysis is completed. Periodic motions will be illustrated for verification of analytical predictions. In addition, the relative force distributions along the displacement are illustrated for illustrations of the analytical conditions of motion switchability on the discontinuous boundary.     

Circular flow on signed graphs

The circular flow number Φc(G,σ) of a signed graph (G,σ) is the minimum r for which an orientation of (G,σ) admits a circular r-flow. We prove that the circular flow number of a signed graph (G,σ) is equal to the minimum imbalance ratio of an orientation of (G,σ). We then use this result to prove that if G is 4-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) (as well as Φ(G,σ)) is at most 4. If G is 6-edge-connected and (G,σ) has a nowhere zero flow, then Φc(G,σ) is strictly less than 4.     

A theory for n-dimensional nonlinear dynamics on continuous vector fields

This paper systematically presents a theory for n-dimensional nonlinear dynamics on continuous vector fields. In this paper, a different view to look into the fundamental theory in dynamics is presented. The ideas presented herein are less formal and rigorous in an informal and lively manner. The ideas may give some inspirations in the field of nonlinear dynamics. The concepts of local and global flows are introduced to interpret the complexity of flows in nonlinear dynamic systems. Further, the global tangency and transversality of flows to the separatrix surface in nonlinear dynamical systems are discussed, and the corresponding necessary and sufficient conditions for such global tangency and transversality are presented. The ε-domains of flows in phase space are introduced from the first integral manifold surface. The domain of chaos in nonlinear dynamic systems is also defined, and such a domain is called a chaotic layer or band. The first integral quantity increment is introduced as an important quantity. Based on different reference surfaces, all possible expressions for the first integral quantity increment are given. The stability of equilibriums and periodic flows in nonlinear dynamical systems are discussed through the first integral quantity increment. Compared to the Lyapunov stability conditions, the weak stability conditions for equilibriums and periodic flows are developed. The criteria for resonances in the stochastic and resonant, chaotic layers are developed via the first integral quantity increment. To discuss the complexity of flows in nonlinear dynamical systems, the first integral manifold surface is used as a reference surface to develop the mapping structures of periodic and chaotic flows. The invariant set fragmentation caused by the grazing bifurcation is discussed. The global grazing bifurcation is a key to determine the global transversality to the separatrix. The local grazing bifurcation on the first integral manifold surface in a single domain without separatrix is a mechanism for the transition from one resonant periodic flow to another one. Such a transition may occur through chaos. The global grazing bifurcation on the separatrix surface may imply global chaos. The complexity of the global chaos is measured by invariant sets on the separatrix surface. The invariant set fragmentation of strange attractors on the separatrix surface is central to investigate the complexity of the global chaotic flows in nonlinear dynamical systems. Finally, the theory developed herein is applied to perturbed nonlinear Hamiltonian systems as an example. The global tangency and tranversality of the perturbed Hamiltonian are presented. The first integral quantity increment (or energy increment) for 2n-dimensional perturbed nonlinear Hamiltonian systems is developed. Such an energy increment is used to develop the iterative mapping relation for chaos and periodic motions in nonlinear Hamiltonian systems. Especially, the first integral quantity increment (or energy increment) for two-dimensional perturbed nonlinear Hamiltonian systems is derived, and from the energy increment, the Melnikov function is obtained under a certain perturbation approximation. Because of applying the perturbation approximation, the Melnikov function only can be used for a rough estimate of the energy increment. Such a function cannot be used to determine the global tangency and transversality to the separatrix surface. The global tangency and transversality to the separatrix surface only can be determined by the corresponding necessary and sufficient conditions rather than the first integral quantity increment. Using the first integral quantity increment, limit cycles in two-dimensional nonlinear systems is discussed briefly. The first integral quantity of any n-dimensional nonlinear dynamical system is very crucial to investigate the corresponding nonlinear dynamics. The theory presented in this paper needs to be further developed and to be treated more rigorously in mathematics.