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Computation of integral manifolds for Carathéodory differential equations

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IMA Journal of Numerical Analysis
Year: 2010 | Volume: 30, Issue: 2 | Journal Article |
We derive two numerical approximation schemes for local invariant manifolds of nonautonomous ordinary differential equations (ODEs) that can be measurable in time and Lipschitzian in the spatial variable. Our approach is inspired by the previous work of Jolly & Rosa (2005, IMA J. Numer. Anal., 25, 698–725) on autonomous ODEs and based on the truncated Lyapunov–Perron operators. Both of our methods...Show More

Computation of integral manifolds for Carathéodory differential equations

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Year: 2010 | Volume: 30, Issue: 2 | Journal Article |
A novelty of distributed parameter estimation strategy for a class of nonlinear system with a time-varying parameter is proposed in this paper. The approach relies upon the concepts of invariant manifold and cooperative persistent excitation condition, does not require a priori knowledge of the time-varying parameters. In addition, it is shown that the parameter estimation error is not only bounde...Show More

Hierarchical Organization in Smooth Dynamical Systems

Artificial Life
Year: 2005 | Volume: 11, Issue: 4 | Journal Article |
Cited by: Papers (1)
This article is concerned with defining and characterizing hierarchical structures in smooth dynamical systems. We define transitions between levels in a dynamical hierarchy by smooth projective maps from a phase space on a lower level, with high dimensionality, to a phase space on a higher level, with lower dimensionality. It is required that each level describe a self-contained deterministic dyn...Show More

Hierarchical Organization in Smooth Dynamical Systems

Year: 2005 | Volume: 11, Issue: 4 | Journal Article |
We consider a control problem of the combustion process of a gas mixture with taking into account the consumption of the reagent and oxidizer. In the system, depending on the values of the control parameter, either slow burnout regime or thermal explosion can be observed. It is shown that there is a critical regime separating these two regimes. The value of the control parameter corresponding to t...Show More

On a modified version of ILDM approach: asymptotic analysis based on integral manifolds

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IMA Journal of Applied Mathematics
Year: 2006 | Volume: 71, Issue: 3 | Journal Article |
Using the method of integral (invariant) manifolds, the intrinsic low-dimensional manifolds (ILDM) method is analysed. This is a method for identifying invariant manifolds of a system's slow dynamics and has proven to be an efficient tool in modelling of laminar and turbulent combustion. It allows treating multi-scale systems by revealing their hidden hierarchy and decomposing the system dynamics ...Show More

On a modified version of ILDM approach: asymptotic analysis based on integral manifolds

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Year: 2006 | Volume: 71, Issue: 3 | Journal Article |
We propose an approach to computing two-dimensional unstable and stable manifolds of three-dimensional vector fields. The main idea is to estimate normal direction on each point around the boundary of current loop of manifold and normalize the normal growth rate during a settled time step to counter the disequilibrium in different directions. In order to enhance the reliability of our approach, li...Show More

Comparative analysis of two asymptotic approaches based on integral manifolds

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IMA Journal of Applied Mathematics
Year: 2004 | Volume: 69, Issue: 4 | Journal Article |
Cited by: Papers (1)
A comparative analysis of the two powerful asymptotic methods, ILDM and MIM (intrinsic low‐dimensional manifolds; method of invariant manifold), is presented in the paper. The two methods are based on the general theory of integral manifolds. The ILDM method is able to handle large systems of ODEs, whereas the MIM method treats systems with a limited number of unknown variables. The MIM method all...Show More

Comparative analysis of two asymptotic approaches based on integral manifolds

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Year: 2004 | Volume: 69, Issue: 4 | Journal Article |
In the research of underactuated control, there are limited results that are for the controlled object with three and more inputs and are based on an invariant manifold theory. For example, there is a switching control method based on an invariant manifold. However, such a control method generates sudden changes of inputs when switching the controllers. In this paper, a control method is proposed ...Show More

Approximation to invariant manifolds under pseudo-hyperbolicity

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IMA Journal of Applied Mathematics
Year: 2016 | Volume: 81, Issue: 1 | Journal Article |
As known from the theory of centre manifolds, it is important to approximate invariant manifolds when discussing high order terms of the reduced systems in the investigation of bifurcations. In this paper, we give a method to approximate invariant manifolds which are obtained under pseudo-hyperbolicity. This method enables us to avoid the trouble of increasing dimension of the so-called approximat...Show More

Approximation to invariant manifolds under pseudo-hyperbolicity

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Year: 2016 | Volume: 81, Issue: 1 | Journal Article |
Control approaches for nonholonomic systems have utilized canonical forms. A nonholonomic double integrator model is the one of canonical forms for nonholonomic systems. Outputs of the controller based on the nonholonomic double integrator model are velocity commands for the kinematic model of a nonholonomic system. In this paper, we consider an extended nonholonomic double integrator model, in wh...Show More
The theory of three-dimensional manifold combination topology is an important branch of low-dimensional topology. Heegaard decomposition is one of the important methods in the topology of three-dimensional manifolds. Casson and Gordon introduced the idea of weakly reducible Heegaard decomposition in 1987, which greatly promoted the research process of Heegaard decomposition theory in combinatorial...Show More

Computation of non-smooth local centre manifolds

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IMA Journal of Numerical Analysis
Year: 2005 | Volume: 25, Issue: 4 | Journal Article |
An iterative Lyapunov–Perron algorithm for the computation of inertial manifolds is adapted for centre manifolds and applied to two test problems. The first application is to compute a known non-smooth manifold (once, but not twice differentiable), where a Taylor expansion is not possible. The second is to a smooth manifold arising in a porous medium problem, where rigorous error estimates are com...Show More

Computation of non-smooth local centre manifolds

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Year: 2005 | Volume: 25, Issue: 4 | Journal Article |
We study the H∞ control problem for an affine singularly perturbed system, which is nonlinear not only in the slow variable (as in the standard case), but also in the fast variable. We construct an e-independent composite controller by solving a slow partial differential equation, that corresponds to the reduced Hamiltonian system, and by solving a fast partial differential equation. The composite...Show More
In this paper an operable, universal and simple theory on the attractiveness of the invariant manifolds of the two-dimensional dynamical systems is first obtained. It is motivated by the Lyapunovdirect method. It means that for any point x→ in the invariant manifold M, n(x→) is the normal passing by x→, and ∀x→ ∈n(x→), if the tangent f(x→) of the orbit of the dynamical system intersects at obtuse ...Show More
Many practical approximations in science and engineering invoke a relatively long physical domain with a relatively thin cross-section. In this scenario, we typically expect the system to have structures that vary slowly in the long dimension. Extant approximation methodologies are typically either self-consistency or limits as the aspect ratio becomes unphysically infinite. The proposed new appro...Show More
We present a novel framework for detection, tracking and recognition of deformable objects undergoing geometric and radiometric transformations. Assuming the geometric deformations an object undergoes, belong to some finite dimensional family, it has been shown that the universal manifold embedding (UME) provides a set of nonlinear operators that universally maps each of the different manifolds, w...Show More
The actor-critic scheme stands for a powerful algorithm to design controllers for linear and non-linear systems subject to changing or highly uncertain dynamics. In particular, the actor-critic scheme that has succeeded is typically based on two neural network stages in a hierarchical architecture where the critic stage approximates the reward cost function. In contrast, the dynamic of the system ...Show More
Due to the relative position invariance of L2 Libration point in the Earth-moon system, Halo orbits around LL2 adapt to the best mission trajectory for Lunar Relay Satellite, which can offer relay communication for space missions such as deep-space exploration and Moon outpost construction. In order to improve the payload launch capacity of Lunar Relay Satellite, the electric thrusters which have ...Show More
The detection of asteroids in near-Earth orbit and celestial bodies in the planetary belt has become one of the important contents of deep space exploration plan. This paper studies the orbital characteristics of probe targeting the asteroid(469219). The libration points (LP) of Earth-asteroid system are calculated, periodic Halo orbit near LP and the invariant manifold properties derived from the...Show More
During the optimal design process of low-thrust transfer trajectory to translunar Libration point, the governing variable such as pitching and yaw angle of satellite were considered. However, the consideration is not enough because the selection of initial orbit after satellite-rocket separation and the inlet point of invariant manifolds are also of great importance to the transition time and the ...Show More
The Sun-Earth L1 point manifolds have many advantages over L2 point manifolds when being used in the design of Earth-Moon transfer trajectory. First, the bi-circular model is introduced, the transformation of state vector between synodic coordinates is given, by choosing a suitable Poincare section, Sun-Earth L1 point manifold and Earth-Moon L2 point manifold are patched together. In order to mini...Show More
This paper studies the design and optimization method of Moon-Mars transfer trajectory with mixed propulsion technology. Previous researches imply that dividing the mission into three phases according to different three body dynamic system the explorer suffers sequentially can facilitate this design problem. Halo orbits about the Earth-Moon L2 and the Mars-Sun L1 points are chosen to insert to the...Show More
The invariant manifold structures of the libration points for restricted three-body problem will provide the framework for understanding complex dynamical phenomena. These stable and unstable invariant manifold tubes associated to libration point orbits can be used to construct new low-energy spacecraft trajectories. So, in this paper, we will introduce the libration point and its invariant manifo...Show More
The importance of parasitic parameters in the study of a dynamical system is brought out in this work. The neglect of these parameters in the context of invariant manifolds for a multi machine power system for transient stability is studied. Singular perturbation approach is used for the calculation of critical clearing time of a large disturbance fault using equilibrium and invariant manifolds. I...Show More
In this work, we consider the state estimation problem for a class of non-autonomous Persidskii systems. This paper presents conditions on the existence and stability of a nonlinear observer based on the invariant manifold approach. The conditions are formulated using Linear Matrix Equalities (LME) and Inequalities (LMI). Two interesting applications of the result are presented: a reduced-order ob...Show More