Stick motions and grazing flows in an inclined impact oscillator

In this paper, the dynamics of an inclined impact oscillator under periodic excitation are investigated using the flow switchability theory of the discontinuous dynamical systems. Different domains and boundaries for such system are defined according to the impact discontinuity. Based on above domains and boundaries, the analytical conditions of the stick motions and grazing motions for the inclined impact oscillator are obtained mathematically, from which it can be seen that such oscillator has more complicated and rich dynamical behaviors. The numerical simulations are given to illustrate the analytical results of complex motions, and several period-1 motions period-2 motion and chaotic motion of the ball in the inclined impact oscillator are also presented. There are more theories about such impact pair to be discussed in future.     

The principle of least constraint for systems with non-restoring constraints

Two new versions of the principle of least constraint are derived from the D'Alembert-Lagrange principle for systems with ideal holonomic and non-holonomic restoring and non-restoring constraints. The first version is similar to Boltzmann's and Bolotov's modification of Gauss's principle for systems with non-restoring constraints. The difference is that here the actual motion is determined in a certain bounded set of possible motions as the one that deviates least from the motion of the system with all non-restoring constraints and any part of the restoring constraints disengaged. According to the second version of the principle, the actual motion is found by comparing certain distinguished possible motions as to their deviation from the motion of the system obtained by eliminating any part of the non-restoring and any part of the restoring constraints. Examples are given.     

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.     

Model-based control strategies for systems with constraints of the program type

The paper presents a model-based tracking control strategy for constrained mechanical systems. Constraints we consider can be material and non-material ones referred to as program constraints. The program constraint equations represent tasks put upon system motions and they can be differential equations of orders higher than one or two, and be non-integrable. The tracking control strategy relies upon two dynamic models: a reference model, which is a dynamic model of a system with arbitrary order differential constraints and a dynamic control model. The reference model serves as a motion planner, which generates inputs to the dynamic control model. It is based upon a generalized program motion equations (GPME) method. The method enables to combine material and program constraints and merge them both into the motion equations. Lagrange’s equations with multipliers are the peculiar case of the GPME, since they can be applied to systems with constraints of first orders. Our tracking strategy referred to as a model reference program motion tracking control strategy enables tracking of any program motion predefined by the program constraints. It extends the “trajectory tracking” to the “program motion tracking”. We also demonstrate that our tracking strategy can be extended to a hybrid program motion/force tracking.     

Modeling and sensitivity analysis methodology for hybrid dynamical system

This paper provides a complete mathematical framework to compute the sensitivities with respect to system parameters for any second order hybrid Ordinary Differential Equation (ODE) and ranked 1 and 3 Differential Algebraic Equation (DAE) system. The hybrid system is characterized by discontinuities in the velocity state variables due to an impulsive forces at the time of event. Such system may also exhibit at the time of event a change in the equation of motions, or in the kinematic constraints.The methodology and the tools developed in this study compute the sensitivities of the states of the model and of the general cost functionals with respect to model parameters for both, unconstrained and constrained, hybrid mechanical systems. The analytical methodology that solves this problem is structured based on jumping conditions for both, the velocity state variables and the sensitivities matrix. The proposed analytical approach is then benchmarked against a known numerical method.Finally, this study emphasizes the penalty formulation for modeling constrained mechanical systems since this formalism has the advantage that it incorporates the kinematic constraints inside the equation of motion, thus easing the numerical integration, works well with redundant constraints, and avoids kinematic bifurcations.     

Periodic and chaotic synchronizations of two distinct dynamical systems under sinusoidal constraints

In this paper, periodic and chaotic synchronizations between two distinct dynamical systems under specific constraints are investigated from the theory of discontinuous dynamical systems. The analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator were obtained, and the invariant domain of sinusoidal synchronization is achieved. From analytical conditions, the control parameter map is developed. Numerical illustrations for partial and full sinusoidal synchronizations of chaotic and periodic motions of the controlled pendulum with the Duffing oscillator are carried out. This paper presents how to apply the theory of discontinuous dynamical systems to dynamical system synchronization with specific constraints. The function synchronization of two distinct dynamical systems with specific constraints should be identified only by G-functions. The significance of function synchronization of distinct dynamical systems is to make the synchronicity behaviors hidden, which is very useful for telecommunication synchronization and network security.     

A <Emphasis Type="Italic">d</Emphasis>-person Differential Game with State Space Constraints

We consider a network of d companies (insurance companies, for example) operating under a treaty to diversify risk. Internal and external borrowing are allowed to avert ruin of any member of the network. The amount borrowed to prevent ruin is viewed upon as control. Repayment of these loans entails a control cost in addition to the usual costs. Each company tries to minimize its repayment liability. This leads to a d -person differential game with state space constraints. If the companies are also in possible competition a Nash equilibrium is sought. Otherwise a utopian equilibrium is more appropriate. The corresponding systems of HJB equations and boundary conditions are derived. In the case of Nash equilibrium, the Hamiltonian can be discontinuous; there are d interlinked control problems with state constraints; each value function is a constrained viscosity solution to the appropriate discontinuous HJB equation. Uniqueness does not hold in general in this case. In the case of utopian equilibrium, each value function turns out to be the unique constrained viscosity solution to the appropriate HJB equation. Connection with Skorokhod problem is briefly discussed.     

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.     

Unifying theory for stability of continuous,discontinuous, and discrete-time dynamical systems

Continuous-time dynamical systems whose motions are continuous with respect to time (called continuous dynamical systems), may be viewed as special cases of continuous-time dynamical systems whose motions are not necessarily continuous with respect to time (called discontinuous dynamical systems, or DDS). We show that the classical Lyapunov stability results for continuous dynamical systems are embedded in the authors’ stability results for DDS (given in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474]), in the following sense: if the hypotheses for a given Lyapunov stability result for continuous dynamical systems are satisfied, then the hypotheses of the corresponding stability result for DDS are also satisfied. This shows that the stability results for DDS in [H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than was previously known, and that the quality of the DDS results therein is consistent with that of the classical Lyapunov stability results for continuous dynamical systems.By embedding discrete-time dynamical systems into a class of DDS with equivalent stability properties, we also show that when the hypotheses of the classical Lyapunov stability results for discrete-time dynamical systems are satisfied, then the hypotheses of the corresponding DDS stability results are also satisfied. This shows that the results for DDS in [H. Ye, A.N. Michel, L. Hou Stability theory for hybrid dynamical systems, IEEE Transactions on Automatic Control 43 (4) (1998) 461–474] are much more general than previously known, having connections even with discrete-time dynamical systems!Finally, we demonstrate by the means of a specific example that the stability results for DDS are less conservative than corresponding classical Lyapunov stability results for continuous dynamical systems.     

Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

On motions and switchability in a periodically forced,discontinuous system with a parabolic boundary

The analytical conditions for motion switchability on the switching boundary in a periodically forced, discontinuous system are developed through the G-function of the vector fields to the switching boundary. Periodic motions in such a discontinuous dynamical system are discussed by the use of mapping structures. Two periodic motions and the analytical conditions are presented for illustration. Further investigation should be carried out for a better understanding of the vanishing and stability of regular and chaotic motions.     

Grazing phenomena in a periodically forced,friction-induced,linear oscillator

The criterion for grazing motions in a dry-friction oscillator is obtained from the local theory of non-smooth dynamical systems on the connectable and accessible domains. The generic mappings for such a dry-friction oscillator are also introduced. The sufficient and necessary conditions for grazing at the final states of mappings are expressed. The initial and final switching sets of grazing mapping, varying with system parameters, are illustrated for the grazing parametric characteristics. The initial and grazing, switching manifolds in the switching sets are defined through grazing mappings. Finally, numerical illustrations of grazing motions are very easily carried out with help of the analytical predictions. This paper provides a comprehensive investigation of grazing motions in the dry-friction oscillator for a better understanding of the grazing mechanism of such a discontinuous system. The investigation based on the local singularity theory is more intuitive and efficient than the discontinuous mapping techniques.     

Coalescing systems of non-Brownian particles

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of Brownian motions on the Sierpinski gasket and stable processes on the real line with stable index greater than one.     

Dynamics of an impact-progressive system

An impact oscillator with a frictional slider is considered. The basic function of the investigated system is to overcome the frictional force and move downwards. Based on the analysis of the oscillatory and progressive motions of the system, we introduce an impact Poincaré map with dynamical variables defined at the impact instants. The nonlinear dynamics of the impact system with a frictional slider is analyzed by using the impact Poincaré map. The stability and bifurcations of single-impact periodic motions are analyzed, and some information about the existence of other types of periodic-impact motions is provided. Since the system equilibrium is moving downwards, one way to monitor the progression rate is to calculate its progression in a finite time. The simulation results show that in a finite time, the largest progression of the system is found to occur for period-1 multi-impact motions existing in the regions of low forcing frequencies. Secondly, the progression of the period-1 single-impact motion with peak-impact velocity is also distinct enough. However, it is important to note, that the largest progression for period-1 multi-impact motion existing at a low forcing frequency is not an optimal choice for practical engineering applications. The greater the number of the impacts in an excitation period, the more distinct the adverse effects such as high noise levels and wear and tear caused by impacts. As a result, the progression of the period-1 single-impact motion with the peak-impact velocity is still optimal for practical applications. The influence of parameter variations on the oscillatory and progressive motions of the impact-progressive system are elucidated accordingly, and feasible parameter regions are provided.     

Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Flows on Homogeneous Spaces

We consider a class of dynamical systems on a compact Lie group G with a left-invariant metric and right-invariant nonholonomic constraints (so-called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G \to Q = G/H, H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. We study the case G = SO(n), when LR systems are ultidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V(k, n) as the corresponding homogeneous spaces. For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere S^n-1. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).     

On nonsmooth and discontinuous problems of stochastic systems optimization

A class of stochastic optimization problems is analyzed that cannot be solved by deterministic and standard stochastic approximation methods. We consider risk-control problems, optimization of stochastic networks and discrete event systems, screening irreversible changes, and pollution control. The results of Ermoliev et al. are extended to the case of stochastic systems and general constraints. It is shown that the concept of stochastic mollifier gradient leads to easily implementable computational procedures for systems with Lipschitz and discontinuous objective functions. New optimality conditions are formulated for designing stochastic search procedures for constrained optimization of discontinuous systems.     

Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.     

Integrate-and-fire models with an almost periodic input function

We investigate leaky integrate-and-fire models (LIF models for short) driven by Stepanov and μ-almost periodic functions. Special attention is paid to the properties of the firing map and its displacement, which give information about the spiking behavior of the considered system. We provide conditions under which such maps are well-defined and are uniformly continuous. We show that the LIF models with Stepanov almost periodic inputs have uniformly almost periodic displacements. We also show that in the case of μ-almost periodic drives it may happen that the displacement map is uniformly continuous, but is not μ-almost periodic (and thus cannot be Stepanov or uniformly almost periodic). By allowing discontinuous inputs, we extend some previous results, showing, for example, that the firing rate for the LIF models with Stepanov almost periodic input exists and is unique. This is a starting point for the investigation of the dynamics of almost-periodically driven integrate-and-fire systems.     

The synthesis of inertial controls for non-stationary systems

The problem of an admissible synthesis of inertia controls for non-stationary systems with a multidimensional control with geometrical constraints on the control and its derivatives is considered. The problem is solved analytically for a linear system: connstructive structure of a family of controls is given, each of which solves the problem, the time of motion from the initial point at zero is calculated and the corresponding trajectory is found. For a non-linear system the problem is solved to a first approximation in the case when there are constraints on the control and on its derivatives.