3,799 research outputs found
Renormalization Group Theory And Variational Calculations For Propagating Fronts
We study the propagation of uniformly translating fronts into a linearly
unstable state, both analytically and numerically. We introduce a perturbative
renormalization group (RG) approach to compute the change in the propagation
speed when the fronts are perturbed by structural modification of their
governing equations. This approach is successful when the fronts are
structurally stable, and allows us to select uniquely the (numerical)
experimentally observable propagation speed. For convenience and completeness,
the structural stability argument is also briefly described. We point out that
the solvability condition widely used in studying dynamics of nonequilibrium
systems is equivalent to the assumption of physical renormalizability. We also
implement a variational principle, due to Hadeler and Rothe, which provides a
very good upper bound and, in some cases, even exact results on the propagation
speeds, and which identifies the transition from ` linear'- to `
nonlinear-marginal-stability' as parameters in the governing equation are
varied.Comment: 34 pages, plain tex with uiucmac.tex. Also available by anonymous ftp
to gijoe.mrl.uiuc.edu (128.174.119.153), file /pub/front_RG.tex (or .ps.Z
On Incremental Stability of Interconnected Switched Systems
In this letter, the incremental stability of interconnected systems is
discussed. In particular, we consider an interconnection of switched nonlinear
systems. The incremental stability of the switched interconnected system is a
stronger property as compared to conventional stability. Guaranteeing such a
notion of stability for an overall interconnected nonlinear system is
challenging, even if individual subsystems are incrementally stable. Here,
preserving incremental stability for the interconnection is ensured with a set
of sufficient conditions. Contraction theory is used as a tool to achieve
incremental convergence. For the feedback interconnection, the small gain
characterisation is presented for the overall system's incremental stability.
The results are derived for a special case of feedback, i.e., cascade
interconnection. Matrix-measure-based conditions are presented, which are
computationally tractable. Two numerical examples are demonstrated and
supported with simulation results to verify the theoretical claims
Contraction analysis of nonlinear systems and its application
The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents
Unstaggered-staggered solitons in two-component discrete nonlinear Schr\"{o}dinger lattices
We present stable bright solitons built of coupled unstaggered and staggered
components in a symmetric system of two discrete nonlinear Schr\"{o}dinger
(DNLS) equations with the attractive self-phase-modulation (SPM) nonlinearity,
coupled by the repulsive cross-phase-modulation (XPM) interaction. These mixed
modes are of a "symbiotic" type, as each component in isolation may only carry
ordinary unstaggered solitons. The results are obtained in an analytical form,
using the variational and Thomas-Fermi approximations (VA and TFA), and the
generalized Vakhitov-Kolokolov (VK) criterion for the evaluation of the
stability. The analytical predictions are verified against numerical results.
Almost all the symbiotic solitons are predicted by the VA quite accurately, and
are stable. Close to a boundary of the existence region of the solitons (which
may feature several connected branches), there are broad solitons which are not
well approximated by the VA, and are unstable
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