Unbounded Trajectories Research Articles

Gauss Equations for Local Action-Angle Orbital Elements in Cislunar Space

To track orbits in cislunar space, predict where they will naturally move over time, and identify where unbounded trajectories come from, one may consider local action-angle orbital elements—coordinates that relate a spacecraft state to specific trajectories and exist in approximately integrable regions of the Earth–moon circular restricted three-body problem (CR3BP). As local action-angle elements are a semi-analytical analog to two-body orbital elements, the theory is extended to allow for the study of arbitrary perturbations to the dynamics. Namely, we derive Gauss equations for an arbitrary perturbing acceleration—continuous or discrete—to the CR3BP. Examples for continuous thrust and impulsive ΔV are provided in cislunar space, i.e., around Earth–moon L1,2 in the CR3BP. Strategies for instantaneous maneuver design and transfers between quasi-periodic orbits and manifolds are developed.

Bubble rising near a vertical wall: Experimental characterization of paths and velocity

Trajectories of a single bubble rising in the vicinity of a vertical solid wall are experimentally investigated. Distinct initial wall-bubble distances are considered for three different bubble rising regimes, i.e., rectilinear, planar zigzag, and spiral. The problem is defined by three control parameters, namely, the Galilei number, Ga, the Bond number, Bo, and the initial dimensionless distance between the bubble centroid and the wall, L. We focus on high-Bond numbers, varying L from 1 to 4, and compare the results with the corresponding unbounded case, L→∞. In all cases, the bubble deviates from the expected unbounded trajectory and migrates away from the wall as it rises due to the overpressure generated in the gap between the bubble and the wall. This repulsion is more evident as the initial wall-bubble distance decreases. Moreover, in the planar zigzagging regime, the wall is found to impose a preferential zigzagging plane perpendicular to it when L is small enough. Only slight wall effects are observed in the velocity or the oscillation amplitude and frequency. The wall migration effect is more evident for the planar zigzagging case and less relevant for the rectilinear one. Finally, the influence of the vertical position of the wall is also investigated. When the wall is not present upon release, the bubbles have the expected behavior for the unbounded case and experience the migration only instants before reaching the wall edge. This repulsion is, in general, more substantial than in the initially present wall case.

Vibration of flexible robots: Dynamics and novel synthesis of unbounded trajectories

Flexible Robotic Systems, by and large, are prone to inherent vibration that recreates itself in several modal frequencies. This in-situ vibration in flexible robots or in any such complaint robotic unit becomes tricky so far as the control system architecture is concerned. Thus, customization of the design and firmware of higher-order flexible robots is highly challenging due to its inherent parameters related to real-time vibration. Vibration in flexible robots has been investigated hitherto from the standpoint of frequency & amplitude tuple, sidetracking the important paradigm of looping of the trajectories. This work has added a technological niche in bringing out the intrinsic dynamics of this vibration from a mathematical perspective of trajectory formation so as to understand the mechanics of spiraling loops while a flexible/compliant robotic system is vibrating under natural conditions. The analytical modeling of the said in-situ vibration has been experimented with through an indigenous single-link flexible robot, fitted with a miniature gripper.

Dynamical and statistical properties of estimated high-dimensional ODE models: The case of the Lorenz '05 type II model.

The performance of estimated models is often evaluated in terms of their predictive capability. In this study, we investigate another important aspect of estimated model evaluation: the disparity between the statistical and dynamical properties of estimated models and their source system. Specifically, we focus on estimated models obtained via the regression method, sparse identification of nonlinear dynamics (SINDy), one of the promising algorithms for determining equations of motion from time series of dynamical systems. We chose our data source dynamical system to be a higher-dimensional instance of the Lorenz 2005 type II model, an important meteorological toy model. We examine how the dynamical and statistical properties of the estimated models are affected by the standard deviation of white Gaussian noise added to the numerical data on which the estimated models were fitted. Our results show that the dynamical properties of the estimated models match those of the source system reasonably well within a range of data-added noise levels, where the estimated models do not generate divergent (unbounded) trajectories. Additionally, we find that the dynamics of the estimated models become increasingly less chaotic as the data-added noise level increases. We also perform a variance analysis of the (SINDy) estimated model's free parameters, revealing strong correlations between parameters belonging to the same component of the estimated model's ordinary differential equation.

Growth of Sobolev norms for coupled lowest Landau level equations

We study coupled systems of nonlinear lowest Landau level equations, for which we prove global existence results with polynomial bounds on the possible growth of Sobolev norms of the solutions. We also exhibit explicit unbounded trajectories which show that these bounds are optimal.

Abnormal route to aging transition in a network of coupled oscillators.

In this article, we investigate the dynamical robustness in a network of Van der Pol oscillators. In particular, we consider a network of diffusively coupled Van der Pol oscillators to explore the aging transition phenomena. Our investigation reveals that the route to aging transition in a network of Van der Pol oscillator is different from that of typical sinusoidal oscillators such as Stuart-Landau oscillators. Unlike sinusoidal oscillators, the order parameter does not follow smooth second-order phase transition. Rather, we observe an abnormal phase transition of the order parameter due to the sudden appearance of unbounded trajectories at a critical point. We provide detailed bifurcation analysis of such an abnormal phase transition. We show that the boundary crisis of a limit-cycle oscillator is at the helm of such an unusual discontinuous path of aging transition.

Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach

The theory of point vortices is used to explain the interaction of a surface vortex with subsurface vortices in the framework of a three-layer quasigeostrophic model. Theory and numerical experiments are used to calculate the interaction between one surface and one subsurface vortex. Then, the configuration with one surface vortex and two subsurface vortices of equal and opposite vorticities (a subsurface vortex dipole) is considered. Numerical experiments show that the self-propelling dipole can either be captured by the surface vortex, move in its vicinity, or finally be completely ejected on an unbounded trajectory. Asymmetric dipoles make loop-like motions and remain in the vicinity of the surface vortex. This model can help interpret the motions of Lagrangian floats at various depths in the ocean.

Decentralized differential drag based control of nanosatellites swarm spatial distribution using magnetorquers

Nanosatellites in the swarm initially move along arbitrary unbounded relative trajectories according to the launch initial conditions. Control algorithms developed in the paper are aimed to achieve the required spatial distribution of satellites in the along-track direction. The paper considers a swarm of 3U CubeSats in LEO, their form-factor is suitable for the aerodynamic control since the ratio of the satellite maximum to minimum cross-section areas is 3. Each satellite is provided with the information about the relative motion of neighboring satellites inside a specified communication area. The paper develops the corresponding decentralized control algorithms using the differential drag force. The required attitude control for each satellite is implemented by the active magnetic attitude control system. A set of decentralized control strategies is proposed taking into account the communicational constraints. The performance of these strategies is studied numerically. The swarm separation effect is demonstrated and investigated.

Multiparticle dynamics on the triangular lattice in interacting media

We study the motion of N particles moving on a two-dimensional triangular lattice, whose sites are occupied by either left or right rotators. These rotators deterministically scatter the particles to the left (right), changing orientation from left to right (right–left) after scattering a particle. This interplay between the scatterers and the particle’s motion causes a single particle to propagate in one direction away from its initial position irrespective of the system’s configuration of scatterers, i.e. the state of the medium through which the particle moves. For multiple particles we show that the particles’ dynamics can be vastly different. Specifically, we show that a particle can become entangled with another particle potentially causing the particle’s trajectory to become periodic and that this can happen when the particles have the same or differing speeds. We describe two classes of periodic orbits based on the particles’ initial velocities. We also describe how a particle with an unbounded past trajectory implies that some, possibly other, particle(s) has an unbounded future trajectory in this and other related multiparticle models.

Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise

Here we study the problem of boundary feedback stabilization to unbounded trajectories for semi-linear stochastic heat equation with cubic non-linearity. The feedback controller is linear, given in a simple explicit form and involves only the eigenfunctions of the Laplace operator. It is supported in a given open subset of the boundary of the domain. Via a rescaling argument, we transform the stochastic equation into a random deterministic one. The simple-form feedback allows to write the solution, of the random equation, in a mild formulation via a kernel. Appealing to a fixed point argument its stability is proved. The approach requires the initial data to be a random variable implying the fact that the solution of the random equation is not adapted. Thus, one cannot recover the solution of the initial stochastic equation from the random one. Hence, the designed feedback controller stabilizes the associated random equation and not the original stochastic equation. Anyway, it stabilizes its random version.

Synchronization of Nonidentical Neural Networks With Unknown Parameters and Diffusion Effects via Robust Adaptive Control Techniques.

This paper considers the self-synchronization and tracking synchronization issues for a class of nonidentically coupled neural networks model with unknown parameters and diffusion effects. Using the special structure of neural networks with global Lipschitz activation function, nonidentical terms are treated as external disturbances, which can then be compensated via robust adaptive control techniques. For the case where no common reference trajectory is given in advance, a distributed adaptive controller is proposed to drive the synchronization error to an adjustable bounded area. For the case where a reference trajectory is predesigned, two distributed adaptive controllers are proposed, respectively, to address the tracking synchronization problem with bounded and unbounded reference trajectories, different decomposition methods are given to extract the heterogeneous characteristics. To avoid the appearance of global information, such as the spectrum of the coupling matrix, corresponding adaptive designs on coupling strengths are also provided for both cases. Moreover, the upper bounds of the final synchronization errors can be gradually adjusted according to the parameters of the adaptive designs. Finally, numerical examples are given to test the effectiveness of the control algorithms.

Oscillation and Extinction Scenarios in the New Continuous Model of the Eruptive Phase of Alien Species Invasion

The paper considers the issue of modeling the development of those special population processes that include the passage of the eruptive phase of dynamics. Such brief hurricane regimes of change are often associated with the consequences of invasions of undesirable species. Processes in the introduction of a species can often develop through the delayed phase of a rapid increase in its abundance. The completion of the phase depends on many factors. Outbreaks of many species exert such a strong pressure on the environment that achieving a non-zero balance equilibrium is problematic. Such phenomena are interpreted by us as an extreme transition process to an uncertain state of the biotic environment before the beginning of the process. Depending on the counteraction, which is clearly seen in the examples of the dynamics of insect pests, simulated scenarios of similar phenomena can develop in various ways, including destruction of the habitat. The new model based on the equation with a deviating argument describes the variant of developing the repeated flash of catastrophic character. The scenario is implemented when non-harmonic cycle N*(rτ, t) occurs, which can not be orbitally stable under the given conditions, but becomes transitive. The cycle ends with the trivial-zero value. The scenario of the most abrupt form of the eruptive phase that we simulate ends in a computational experiment by the death of the invasive population, but without forming an unbounded trajectory from the oscillations, as was the case with the destruction of the relaxation cycle of the extreme amplitude in the population flash equation in our previous work.

Chaotic backward shift operator on Chebyshev polynomials

We obtain the representation of the backward shift operator on Chebyshev polynomials involving a principal value (PV) integral. Twice the backward shift on the space of square-summable sequences l2 displays chaotic dynamics, thus we provide an explicit form of a chaotic operator on L2 (−1, 1, (1−x2)–1/2) using Cauchy’s PV integral. We explicitly calculate the periodic points of the operator and provide examples of unbounded trajectories, as well as chaotic ones. Histograms and recurrence plots of shifts of random Chebyshev expansions display interesting behaviour over fractal measures.

Early models of chemical oscillations failed to provide bounded solutions.

Before the development of the Brusselator, the Sel'kov and Turing models were considered as possible prototype models of chemical oscillations. We first analyse their Hopf bifurcation branches and show that they become vertical past a critical value of the control parameter. We explain this phenomenon by the emergence of canard orbits. Second, we analyse all solutions in the phase plane and show that some initial conditions lead to unbounded trajectories even if there exists a locally stable attractor. Our findings mathematically explain why these too simple two-variable models fail to account for the emergence of chemical oscillations. They support the conclusion that the Brusselator is the first minimal two-variable model explaining the onset of stable oscillations in a way fully compatible with thermodynamics and the law of mass action.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Threshold speed for two-dimensional confinement of charged particles in certain axisymmetric magnetic fields

In this paper we discuss conditions under which charged particles are confined by an axisymmetric longitudinal magnetic field with power law dependence on the radius. We derive a transcendental equation for the critical speed corresponding to the threshold between bounded and unbounded trajectories of the particles. This threshold speed shows strong dependence on the direction, and this dependence becomes more prominent as the exponent of the power law increases. The equation for threshold speed can be solved exactly for several specific values of the power exponent, but in general it requires a numerical treatment. Remarkably, if the magnetic field magnitude decreases more slowly than the inverse of the radius, charged particles remain confined no matter how large their energies may be.

Dynamical analysis of an n−H−T cosmological quintessence real gas model with a general equation of state

The cosmological dynamics of a quintessence model based on real gas with general equation of state is presented within the framework of a three-dimensional dynamical system describing the time evolution of the number density, the Hubble parameter and the temperature. Two global first integrals are found and examples for gas with virial expansion and van der Waals gas are presented. The van der Waals system is completely integrable. In addition to the unbounded trajectories, stemming from the presence of the conserved quantities, stable periodic solutions (closed orbits) also exist under certain conditions and these represent models of a cyclic Universe. The cyclic solutions exhibit regions characterized by inflation and deflation, while the open trajectories are characterized by inflation in a “fly-by” near an unstable critical point.

Output regulation in the presence of quadratically bounded parameter uncertainties

The problem of robust output regulation is studied for a class of parameter uncertain systems under unity output feedback control. The objective is tracking of the desired reference trajectory in the presence of the disturbance, both generated by a common exosystem. Because of the anti-stability of the exosystem and potentially unbounded reference trajectory, the output tracking error is employed as the measurement signal and used as the input to the feedback controller. The method of p-copy of the internal model is utilised to augment the plant dynamics. Assuming that the output regulation condition is satisfied for all the parameter uncertainties, it is shown that the problem of robust output regulation is equivalent to the problem of robust output stabilisation. Furthermore, for quadratically bounded parameter uncertainties, an application of the notion of the quadratic stability leads to H ∞ -based robust control, and the maximum allowable uncertainty bound can be computed, below which the robust output regulation can be achieved.

Output Regulation for a Class of Parameter Uncertain Systems

The problem of robust output regulation is studied for a class of parameter uncertain systems under unity output feedback control. The objective is tracking of the desired reference trajectory in the presence of the disturbance, both generated by a common exosystem. Because of the anti-stability of the exosystem and potentially unbounded reference trajectory, the output tracking error is employed as the measurement signal and used as the input to the feedback controller. The method of p-copy of the internal model is utilized to augment the plant dynamics. Assuming that the output regulation condition is satisfied for all the parameter uncertainties, it is shown that the problem of robust output regulation is equivalent to the problem of robust output stabilization. Furthermore for quadratically bounded parameter uncertainties, an application of the notion of the quadratic stability leads to H∞ based robust control, and the maximum allowable uncertainty bound can be computed, below which the robust output regulation can be achieved.

Nonlinear dynamics of an elliptic vortex embedded in an oscillatory shear flow.

The nonlinear dynamics of an elliptic vortex subjected to a time-periodic linear external shear flow is studied numerically. Making use of the ideas from the theory of nonlinear resonance overlaps, the study focuses on the appearance of chaotic regimes in the ellipse dynamics. When the superimposed flow is stationary, two general types of the steady-state phase portrait are considered: one that features a homoclinic separatrix delineating bounded and unbounded phase trajectories and one without a separatrix (all the phase trajectories are bounded in a periodic domain). When the external flow is time-periodic, the ensuing nonlinear dynamics differs significantly in both cases. For the case with a separatrix and two distinct types of phase trajectories: bounded and unbounded, the effect of the most influential nonlinear resonance with the winding number of 1:1 is analyzed in detail. Namely, the process of occupying the central stability region associated with the steady-state elliptic critical point by the stability region associated with the nonlinear resonance of 1:1 as the perturbation frequency gradually varies is investigated. A stark increase in the persistence of the central regular dynamics region against perturbation when the resonance of 1:1 associated stability region occupies the region associated with the steady-state elliptic critical point is observed. An analogous persistence of the regular motion occurs for higher perturbation frequencies when the corresponding stability islands reach the central stability region associated with the steady-state elliptic point. An analysis for the case with the resonance of 1:2 is presented. For the second case with only bounded phase trajectories and, therefore, no separatrix, the appearance of much bigger stability islands associated with nonlinear resonances compared with the case with a separatrix is reported.

Complexity in Linear Systems: A Chaotic Linear Operator on the Space of Odd2π-Periodic Functions

Not just nonlinear systems but infinite-dimensional linear systems can exhibit complex behavior. It has long been known that twice the backward shift on the space of square-summable sequencesl2displays chaotic dynamics. Here we construct the corresponding operatorCon the space of2π-periodic odd functions and provide its representation involving a Principal Value Integral. We explicitly calculate the eigenfunction of this operator, as well as its periodic points. We also provide examples of chaotic and unbounded trajectories ofC.