Region Of Initial Conditions Research Articles

On precession of Lagrange’s top

The article describes the results obtained for the upper and lower bounds (estimates) for the apsidal angle (precession angle) in the theory of the motion of the heavy symmetrical solid body about fixed point (Lagrange’s case) for arbitrary initial conditions and parameters of the body. All regions of initial conditions is divided into two sets. In the first set there is a direct precession of the top, in the second set there is a retrograde precession of the top.

On a discrete model of tumour–immune system interactions with blockade of immune checkpoints

A discrete-time model of tumour–immune system interactions based on a published model is derived. The tumour is assumed to grow exponentially while the immune system can either slow down or shrink tumour growth. However, cancerous tumour cells have many mechanisms to impair the immune system. The mathematical model therefore incorporates immunotherapy of immune checkpoint inhibitors to study tumour dynamics. It is proven that the tumour can grow to unboundedly large if its intrinsic growth rate exceeds a critical value or if its initial size is beyond a threshold when first detected. In addition, a region of initial conditions for which the tumour will eventually reach an unbounded size is derived when the tumour growth rate is smaller than the critical value. As a result, the immunotherapy fails to control the aggressive tumour. There are two positive equilibrium points when the tumour growth rate is smaller than the critical value. We verify that a saddle-node bifurcation occurs at the critical tumour growth rate whereas the equilibrium with the larger tumour size is always unstable. The equilibrium with the smaller tumour size may undergo a subcritical Neimark–Sacker bifurcation in certain parameter regimes. Parameter values derived from various clinical studies are simulated to illustrate our analytical findings.

The inverse problem of stabilization of a spherical pendulum in a given position under oblique vibration

The inverse problem is posed of stabilizing a spherical pendulum (a mass point at the end of a weightless solid rod of length l ) in a given position using high-frequency vibration of the suspension point. The position of the pendulum is determined by the angle between the pendulum rod and the gravity acceleration vector. For any given position of the pendulum, a series of oblique vibration parameters (amplitude of the vibration velocity and the angle between the vibration velocity vector and the vertical) were found that stabilize the pendulum in this position. From the obtained series of solutions, the parameters of optimal vibration (vibration with a minimum amplitude of velocity) are selected depending on the position of the pendulum. The region of initial conditions is studied, of which the optimal vibration leads the pendulum to a predetermined stable position after a sufficiently long time. This area, following N. F.Morozov et al., called the area of attraction.

An Observer-Based Controller for Congestion Control in Data Networks

This paper investigates the design of an observer-based controller to overcome the congestion problem in the TCP/IP networks where the available link bandwidth is modeled as a time-variant disturbance. Then, a detailed description is discussed to establish a linearized mathematical model in order to ensure the robust stability of the saturated delayed system and satisfy the $$H_{\infty }$$ performance. Based on a Lyapunov–Krasovskii (L–K) functional, a generalized sector condition, and the Finsler’s Lemma, the proposed new design methodology leads to a quite simple LMI condition that is numerically tractable with any convex optimization algorithm. An effective iterative optimization algorithm is adopted to estimate the largest possible region of initial conditions as will be seen in the numerical examples where the results are compared to PI, RED, and REM controllers.

Unifying averaged dynamics of the Fokker-Planck equation for Paul traps

Collective dynamics of a collisional plasma in a Paul trap is governed by the Fokker-Planck equation, which is usually assumed to lead to a unique asymptotic time-periodic solution irrespective of the initial plasma distribution. This uniqueness is, however, hard to prove in general due to analytical difficulties. For the case of small damping and diffusion coefficients, we apply averaging theory to a special solution to this problem and show that the averaged dynamics can be represented by a remarkably simple 2D phase portrait, which is independent of the applied rf field amplitude. In particular, in the 2D phase portrait, we have two regions of initial conditions. From one region, all solutions are unbounded. From the other region, all solutions go to a stable fixed point, which represents a unique time-periodic solution of the plasma distribution function, and the boundary between these two is a parabola.

Network-based practical set consensus of multi-agent systems subject to input saturation

This paper is concerned with the network-based practical set consensus problem of multi-agent systems subject to input saturation constraints. Considering network-induced delays, data quantization and aperiodic sampling intervals, a network-based framework which allows each agent to be remotely controlled over the communication network is established. Under this framework, a new network-based consensus protocol with input saturation constraints is proposed. With this protocol, the consensus problem can be transformed into the stabilization problem of time-delay systems with bounded perturbations. By using the Lyapunov–Krasovskii approach, a stability condition guaranteeing that the error system can exponentially converge to a bounded set is derived, where the region of initial conditions can be estimated by considering the effect of the first delay interval. Based on this stability condition, the network-based consensus controller gain matrix can be obtained. An optimization algorithm is introduced for simultaneously designing the consensus controller gain and estimating the region of attraction as small as possible and the region of initial conditions as large as possible. A numerical example is given to illustrate the efficiency of all the results derived in this paper.

New stability and exact observability conditions for semilinear wave equations

The problem of estimating the initial state of 1-D wave equations with globally Lipschitz nonlinearities from boundary measurements on a finite interval was solved recently by using the sequence of forward and backward observers, and deriving the upper bound for exact observability time in terms of Linear Matrix Inequalities (LMIs) (Fridman, 2013). In the present paper, we generalize this result to n-D wave equations on a hypercube. This extension includes new LMI-based exponential stability conditions for n-D wave equations, as well as an upper bound on the minimum exact observability time in terms of LMIs. For 1-D wave equations with locally Lipschitz nonlinearities, we find an estimate on the region of initial conditions that are guaranteed to be uniquely recovered from the measurements. The efficiency of the results is illustrated by numerical examples.

Search for periodic orbits in the general three-body problem

An original method for searching for regions of initial conditions giving rise to close to periodic orbits is proposed in the framework of the general three-body problem with equal masses and zero angular momentum. Until recently, three stable periodic orbits were known: the Schubart orbit for the rectilinear problem, the Broucke orbit for the isosceles problem, and the Moore eight-figure orbit. Recent studies have also found new periodic orbits for this problem. The proposed method minimizes a functional that calculates the sum of squared differences between the initial and current coordinates and the velocities of the bodies. The search is applied to short-period orbits with periodsT < 10τ, where τ is the mean crossing time for the components of the triple system. Twenty one regions of initial conditions, each corresponding to a particular periodic orbit, have been found in the current study. A criterion for the reliability of the results is that the initial conditions for the previously known stable periodic orbits are located inside the regions found. The trajectories of the bodies with the corresponding initial conditions are presented. The dynamics and geometry of the orbits constructed are described.

Real-time capable trajectory synthesis via multivariate interpolation methods for a moon landing manoeuvre

Conventional solutions to control design problems in complex space missions are often based on a solution of an associated optimal control problem. Due to limited onboard computing capacity, the computation of an optimal trajectory and a dedicated controller are commonly carried out offline, in advance of a mission. This approach requires the definition of a nominal set of initial conditions and system parameters, which are generally different from the conditions during the actual manoeuvre. This paper proposes an approach based on multivariate spline interpolation for generating nearly optimal solutions online, despite the difference between the nominal and actual conditions during the manoeuvre. The onboard computations are limited to simple interpolations and hence do not cause considerable additional computing effort. Compared to conventional control design approaches, the proposed method significantly increases the region of initial conditions and size of parametric uncertainties for which the manoeuvre can be successfully completed. The applicability and practical relevance of this approach are demonstrated for a Moon landing manoeuvre.

Approximate solution to the resource consumption minimization problem. II. Estimates for the proximity of controls

The proximity is estimated of a resource quasioptimal control to the optimal control. We give a method for subdividing the bounded region of initial conditions into subregions and bringing the quasioptimal control closer to the resource optimal control.

Bi-fidelity fitting and optimization

A common feature in computations of chemical and physical properties is the investigation of phenomena at different levels of computational accuracy. Less accurate computations are used to provide a relatively quick understanding of the behavior of a system and allow a researcher to focus on regions of initial conditions and parameter space where interesting phenomena are likely to occur. These inexpensive calculations are often discarded when more accurate calculations are performed. This paper demonstrates how computations at different levels of accuracy can be simultaneously incorporated to study chemical and physical phenomena with less overall computational effort than the most expensive level of computation. A smaller set of computationally expensive calculations is needed because the set of expensive calculations is correlated with the larger set of less expensive calculations. We present two applications. First, we demonstrate how potential energy surfaces can be fit by simultaneously using results from two different levels of accuracy in electronic structure calculations. In the second application, we study the optical response of metallic nanostructures. The optical response is generated with calculations at two different grid resolutions, and we demonstrate how using these two levels of computation in a correlated fashion can more efficiently optimize the response.

The bouncing motion of a superball between a horizontal floor and a vertical wall

In earlier work [P.J. Aston, R. Shail, The dynamics of a bouncing superball with spin, Dyn. Sys. 22 (2007) 291–322] the problem of the possible back and forth motion of a superball thrown spinning onto a horizontal plane was considered in detail. In this paper the problem is extended to include a vertical wall. In particular motion of the superball where it bounces alternately on the floor and the wall several times is considered. Using the same physical model as in our previous work, a non-linear mapping is derived which relates the launch data of the (n+1)th floor bounce to that of the n th. This mapping is analysed both numerically and theoretically, and a detailed description is presented of various possible motions. Regions of initial conditions which result in a specified number of bounces against the wall are also considered.

Predator–prey system with strong Allee effect in prey

Global bifurcation analysis of a class of general predator-prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.

The rectilinear three-body problem

The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities, escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches or a definite time. In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this periodic orbit. We have studied the phase space structure via Poincare sections. Using these sections and symbolic dynamics, we study the fine structure of the region of initial conditions, in particular the chaotic scattering region.

Four easy pieces

This paper is a mini-summary of four new advances in the theory of PT-symmetric quantum mechanics. It describes some new calculations that were completed in 2005. The first advance concerns the classical coordinate-space trajectories in some PT-symmetric theories. Depending on the initial conditions, one can find arbitrarily long periodic PT-symmetric classical trajectories. The longer trajectories originate from smaller regions of initial conditions. There is an interesting resemblance to the so-called period-doubling route to chaos. The second advance concerns the perturbative construction of the C operator for a PT-symmetric square well. The result has notable similarities to and differences from that for the PT-symmetric cubic oscillator. The third advance is a detailed comparison of a PT-symmetric Hamiltonian and the corresponding Hermitian Hamiltonian. It is argued that a perturbative construction of the Hermitian Hamiltonian leads to an almost intractable theory. The fourth advance concerns reflectionless potentials and PT symmetry and a possible connection with problems in cosmology.

Edge of Chaos in a Parallel Shear Flow

We study the transition between laminar and turbulent states in a Galerkin representation of a parallel shear flow, where a stable laminar flow and a transient turbulent flow state coexist. The regions of initial conditions where the lifetimes show strong fluctuations and a sensitive dependence on initial conditions are separated from the ones with a smooth variation of lifetimes by an object in phase space which we call the "edge of chaos." We describe techniques to identify and follow the edge, and our results indicate that the edge is a surface. For low Reynolds numbers we find that the surface coincides with the stable manifold of a periodic orbit, whereas at higher Reynolds numbers it is the stable set of a higher-dimensional chaotic object.

Robust output feedback stabilization of uncertain time-varying state-delayed systems with saturating actuators

The robust output feedback stabilization problem for state-delayed systems with time-varying delays and saturating actuators is addressed here. The systems considered are continuous-time, with parametric uncertainties entering all the matrices in the system representation. A saturating control law is designed and a region of initial conditions is specified within which local asymptotic stability of the closed-loop system is ensured. The least conservative approach that employs the Lyapunov–Krasovskii functional is adopted to ensure stabilization. The designed controller is dependent on the time-delay and its rate of change. The controller is constructed in terms of the solution to a set of matrix inequalities.

Load-flow fractals draw clues to erratic behaviour

Fractal images have been discovered in many areas of science and engineering in the past two decades, so it is not surprising that they have also appeared in power system literature. This article provides a brief introduction to fractals and presents some fractal patterns produced by Newton-Raphson load-flow calculations for a small power system. If we imagine performing load-flow calculations from a dense grid of initial conditions, the region of initial conditions that converge to a particular equilibrium point using the Newton-Raphson method is seen to have a fractal boundary.

Initial conditions and dynamics of triple systems

The dynamical evolution of triple systems with equal-mass components and zero initial velocities is studied. We consider two regions of initial conditions: a regionD of all possible configurations of triples and a circleR. The configurations are distributed uniformly within these regions. The calculations have been carried out until a time when escape or ‘conditional’ escape (i.e. distant ejection) of one component takes place. The accuracy has been checked by doing time-reversed integration. Types of ‘predictable’ and ‘non-predictable’ systems are revealed. Averages for a number of evolution parameters are presented: the life-time, minimum perimeter during the last triple approach resulting in escape, semi-major axis and eccentricity of the final binary, and the smallest separation between the components during the evolution. It is shown that the statistical results for the regionsD andR do not differ significantly for the most part. Our results, which have been obtaned by a three-body regularization method, are in good agreement with previous work based on the RK4 integrator and Sundman's time smoothing.

Trapping of particles in a time-dependent Hamiltonian

The trapping of particles in the case of two coupled oscillators, when the frequency of one approaches the constant frequency of the other exponentially in time, is investigated. It is found that we can predict the region of initial conditions for which a test particle can be trapped around a resonant periodic orbit.