Piece-wise System Research Articles

A Predator–Prey Model with Prey Population Guided Anti-Predator Behavior

We consider a predator–prey system with prey population guided anti-predator behavior, in which anti-predator behaviors happen only when the population size of the prey is greater than a threshold. We investigate the rich dynamics of the proposed piecewise model as well as both subsystems without and with nonlinear functional response. In particular, the subsystem with anti-predator behaviors exhibits rich dynamical behaviors including saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Further, besides the dynamical properties of subsystems the piecewise system shows some new complicated dynamical behaviors as the threshold value varies, including unstable limit cycle, semistable limit cycle, bistability of equilibrium and limit cycle, and tristability of three equilibria. From the switching system we can conclude that a great anti-predator rate induces the prey population to persist more likely, but whether the prey and predator populations coexist depends further on the threshold that triggers anti-predator behavior. Especially, a large threshold not only makes coexistence of the prey and predator populations as an equilibrium more likely, but also damps the predator–prey oscillations.

Bifurcation Analysis of Planar Piecewise Linear System with Different Dynamics

In this paper, we investigate the bifurcation phenomena of a planar piecewise linear system. This piecewise linear system comprises two linear subsystems. The two linear subsystems have different types of dynamics. One subsystem has node or saddle dynamic and the other has focus dynamic. Some sufficient and necessary conditions for the existence of periodic orbit are given by studying the properties of Poincaré maps. Our results show that two crossing periodic orbits can bifurcate from this piecewise linear system. Moreover, we establish some sufficient and necessary conditions for the existence of sliding periodic orbit, crossing–sliding periodic orbit and sliding homoclinic orbit passing through a pseudo saddle and so on. We find that this piecewise system can appear multiply as two limit cycle bifurcation, buckling bifurcation, critical crossing cycle bifurcation, sliding homoclinic bifurcation, pseudo homoclinic bifurcation and so on. To our knowledge, sliding bifurcation phenomena are usually ignored when people study piecewise linear systems.

Modelling the regulatory system of a chemostat model with a threshold window

The chemostat model involving a control is either on or off which can be described by piecewise (or non-smooth) system. To improve the regulatory system for microorganism culture, piecewise chemostat models involving control strategy with threshold window are proposed and analysed. A special case is investigated first, i.e., chemostat models with a single threshold, the existence and stability of regular, virtual, pseudo-equilibria and tangent points are addressed, and it is shown that the regular equilibria and the pseudo-equilibrium cannot coexist. Then the global stabilities of the regular equilibria and pseudo-equilibrium have been studied by using qualitative analysis techniques of non-smooth Filippov dynamic systems. Furthermore, sliding bifurcations related to boundary node bifurcations are investigated with theoretical and numerical techniques, and the corresponding biological implications are discussed. For chemostat models with a threshold window, the effects of the relations between the threshold window and equilibria on the global dynamics of the proposed models are indicated, and subsequently we elaborate how the widths of the threshold windows affect the durations of No control state and Control state. All results suggest that the microorganism concentration can be maintained within a prescribed range using the proposed piecewise chemostat models. Moreover, it is concluded that the initial microorganism concentrations play an important role in the control process.

Complexity Analysis of the Piece-Wise Affine Approximation for the Car on the Nonlinear Hill Model Related To Discrete–Time, Minimum Time Control Problem

The main aim of the paper is to analyse effectiveness of the approximation method from the nonlinear discretised into discrete piece-wise affine model. Accuracy and numerical complexity of the piece-wise affine control system grow with the number of polyhedral partitions, that describe the system. The model with state dependent nonlinearity can be effectively approximated by proposed secant piece-wise linear approximation. The effectiveness of the proposed method is evaluated on the car on the nonlinear hill model. DOI: http://dx.doi.org/10.5755/j01.eee.20.10.4454

Characteristic Exponent Extraction of Experimental Nonlinear Time Series

The experiment of chaotic exponent extraction is carried out on the basis of nonlinear vibration system, and the responses under different conditions are processed with the improved algorithm. Firstly, the weighted wavelet denoise method is applied to filter the contaminated noise. Then, on the basis of fast search technology i.e. space grid hiberarchy inquiry method, the chaos characteristic exponent extraction algorithm is modified and applied to LE and fractal dimension calculation. Finally the piece-wise vibration system is designed, and the nonlinear dynamics under different harmonic frequencies excitation are analyzed. The comprehensive chaos judgment program is developed, in which the time domain diagram, phase space reconstruction attractor, Lyapunov exponent, fractal dimension curve of the measured data are obtained. The interesting phenomena such as AM modulation, limited cycle, and strange attractor are observed.

Anti-control of Hopf bifurcation in the new chaotic system with piecewise system and exponential system

In order to understand the complex three-dimensional dynamical system with the unstable nodes, we propose a nonlinear controller. The corresponding controlling system makes the codimension one, two, and three Hopf bifurcations happen. The mathematical deduction demonstrates that the system can be controlled to produce the degenerate Hopf bifurcation at desired location and stability of controllable bifurcation.

Global dynamics of a piece-wise epidemic model with switching vaccination strategy

A piece-wise epidemic model of a switching vaccination program, implemented once the number of people exposed to a disease-causing virus reaches a critical level, is proposed. In addition, variation or uncertainties in interventions are examined with a perturbed system version of the model. We also analyzed the global dynamic behaviors of both the original piece-wise system and the perturbed version theoretically, using generalized Jacobian theory, Lyapunov constants for a non-smooth vector field and a generalization of Dulac's criterion. The main results show that, as the critical value varies, there are three possibilities for stabilization of the piece-wise system: (i) at the disease-free equilibrium; (ii) at the endemic states for the two subsystems or (iii) at a generalized equilibrium which is a novel global attractor for non-smooth systems. The perturbed system exhibits new global attractors including a pseudo-focus of parabolic-parabolic (PP) type, a pseudo-equilibrium and a crossing cycle surrounding a sliding mode region. Our findings demonstrate that an infectious disease can be eradicated either by increasing the vaccination rate or by stabilizing the number of infected individuals at a previously given level, conditional upon a suitable critical level and the parameter values.

In-Flight Alignment UsingH∞Filter for Strapdown INS on Aircraft

In-flight alignment is an effective way to improve the accuracy and speed of initial alignment for strapdown inertial navigation system (INS). During the aircraft flight, strapdown INS alignment was disturbed by lineal and angular movements of the aircraft. To deal with the disturbances in dynamic initial alignment, a novel alignment method for SINS is investigated in this paper. In this method, an initial alignment error model of SINS in the inertial frame is established. The observability of the system is discussed by piece-wise constant system (PWCS) theory and observable degree is computed by the singular value decomposition (SVD) theory. It is demonstrated that the system is completely observable, and all the system state parameters can be estimated by optimal filter. Then a H ∞ filter was designed to resolve the uncertainty of measurement noise. The simulation results demonstrate that the proposed algorithm can reach a better accuracy under the dynamic disturbance condition.

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the well-known Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameter three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems.In the piecewise system, the noose bifurcation involves reversible periodic orbits that intersect the separation plane at two or four points. This work is focused on those reversible periodic orbits that intersect the separation plane twice (RP2-orbits). It is established that for every T between 2π and a critical point, there exists a unique value of the parameter for which the system has an RP2-orbit with period T. Moreover, this critical value, that separates periodic orbits with two or four points of intersection with the separation plane, corresponds to an RP2-orbit that crosses the separation plane tangentially.It is also proved that in a parameter versus period bifurcation diagram, the curve of this family of periodic orbits has a unique maximum point, which corresponds to the saddle-node bifurcation of periodic orbits that appears in the noose bifurcation.

Optimal complexity reduction of polyhedral piecewise affine systems

This paper focuses on the N P -hard problem of reducing the complexity of piecewise polyhedral systems (e.g. polyhedral piecewise affine (PWA) systems). The results are fourfold. Firstly, the paper presents two computationally attractive algorithms for optimal complexity reduction that, under the assumption that the system is defined over the cells of a hyperplane arrangement, derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra. The algorithms are based on the cells and the markings of the hyperplane arrangement. In particular, the first algorithm yields a set of disjoint (non overlapping) merged polyhedra by executing a branch and bound search on the markings of the cells. The second approach leads to non-disjoint (overlapping) polyhedra by formulating and solving an equivalent (and well-studied) logic minimization problem. Secondly, the results are extended to systems defined on general polyhedral partitions (and not on cells of hyperplane arrangements). Thirdly, the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable. Fourthly, the paper shows that based on the notion of the hyperplane arrangement PWA state feedback control laws can be implemented efficiently. Three examples, including a challenging industrial problem, illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude.

Piecewise Optimum Delay Estimation for Improved Signal Control

An adaptive control strategy called Piecewise Optimum Delay Estimation (PODE) was developed and tested. This paper reviews the advantages of the existing adaptive control strategies over the actuated control scheme and uses their limitations as the basis for improvement. By using real-time traffic as input, PODE dynamically searches movement combinations for phasing and timing decisions that minimize piecewise system delay. Preliminary results show that PODE can considerably and consistently reduce delay more than actuated control can. Supported by a self-adjusting procedure, PODE control logic has demonstrated its competitive strength among other adaptive control strategies, especially under congested and oversaturated conditions.

Arbitrary Periodic Motions and Grazing Switching of a Forced Piecewise Linear, Impacting Oscillator

Abstract The grazing bifurcation and periodic motion switching of the harmonically forced, piecewise linear system with impacting are investigated. The generic mappings relative to the discontinuous boundaries of this piecewise system are introduced. Based on such mappings, the corresponding grazing conditions are obtained. The mapping structures are developed for the analytical prediction of periodic motions in such a system. The local stability and bifurcation conditions for specified periodic motions are obtained. The regular and grazing, periodic motions are illustrated. The grazing is the origin of the periodic motion switching for this system. Such a grazing bifurcation cannot be estimated through the local stability analysis. This model is applicable to prediction of periodic motions in nonlinear dynamics of gear transmission systems.

Homoclinic orbits in a piecewise system and their relation with invariant sets

Basic phenomena in chaos can be associated with homoclinic and heteroclinic orbits. In this paper, we present a general numerical method to demonstrate the existence of these orbits in piecewise-linear systems. We also show that the tangency of the stable and unstable manifolds, at the onset of the chaotic double-scroll attractor, changes the basin boundaries of two α-limit sets. These changes are evidence of homoclinicity in the dynamical system. These basins give complete information about the stable manifolds around the fixed points. We show that trajectories that depart from these boundaries (for backward integration) are bounded sets. Moreover, we also show that the unstable manifolds are geometrically similar to the existing attracting sets. In fact, when no homo- (hetero-)clinic orbits exist, the attractors are ω-limit sets of initial conditions on the unstable manifolds.