1D Linear Systems Research Articles

Trapping wave fields in an expulsive potential by means of linear coupling

We demonstrate the existence of confined states in one- and two-dimensional (1D and 2D) systems of two linearly coupled components, with the confining harmonic-oscillator (HO) potential acting upon one component and an expulsive anti-HO potential acting upon the other. The systems can be implemented in optical and BEC dual-core waveguides. In the 1D linear system, codimension-one solutions are found in an exact form for the ground state (GS) and dipole mode (the first excited state). Generic solutions are produced by means of the variational approximation and are found in a numerical form. Exact codimension-one solutions and generic numerical ones are also obtained for the GS and vortex states in the 2D system (the exact solutions are found for all values of the vorticity). Both the trapped and antitrapped components of the bound states may be dominant ones, in terms of the norm. The localized modes may be categorized as bound states in continuum, as they coexist with delocalized ones. The 1D states, as well as the GS in 2D, are weakly affected and remain stable if the self-attractive or repulsive nonlinearity is added to the system. The self-attraction makes the vortex states unstable against splitting, while they remain stable under the action of the self-repulsion.

Push-pull thiophene chromophores for electro-optic applications: from 1D linear to β-branched structures.

We report the synthesis and characterization of a novel series of push-pull chromophores bearing 1D linear and β-branched thiophenes as π-conjugated spacers between a 2,2,4,7-tetramethyl-1,2,3,4-tetrahydroquinoline electron donor unit and dicyano- and tricyanovinylene electron acceptor groups. The effect of the introduction of β-thiophenes on the linear and nonlinear (NLO) optical properties as well as electrochemical and thermal data is studied in detail by performing a comparative study between the branched and 1D linear systems. In addition, a parallel DFT computational study is used to evaluate structure-property relationships. The non-linear optical behavior of the molecules both in solution and in solid state as electro-optic (EO) films using a guest-host approach shows very promising performance for electro-optic applications with high molecular first hyperpolarizabilities (μβ) of 4840 × 10-48 esu and electro-optic coefficients r33 reaching 650 pm V-1. One highlight is that the electro-optic films of the β-branched chromophores are superior in terms of thermal stability in device operation as measured by a transmissive modified reflective Teng-Man method. This work provides guidelines for the design of improved electro-optic materials including β-branched chromophores which could be useful for practical EO applications, where both enhanced β and r33 values together with chemical and thermal stability are necessary.

A New 7D Hyperchaotic System with Five Positive Lyapunov Exponents Coined

This paper reports the finding of a new seven-dimensional (7D) autonomous hyperchaotic system, which is obtained by coupling a 1D linear system and a 6D hyperchaotic system that is constructed by adding two linear feedback controllers and a nonlinear feedback controller to the Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest is that it has a hyperchaotic attractor with five positive Lyapunov exponents and a unique equilibrium in a large range of parameters. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, power spectrum and Poincaré projections verifies the existence of hyperchaotic and chaotic attractors. Moreover, stability of the hyperbolic equilibrium is analyzed and a complete mathematical characterization for 7D Hopf bifurcation is given. Finally, circuit experiment implements the hyperchaotic attractor of the 7D system, showing very good agreement with the simulation results.

A new 5D hyperchaotic system based on modified generalized Lorenz system

This paper reports a new five-dimensional (5D) hyperchaotic system with three positive Lyapunov exponents, which is generated by adding a linear controller to the second equation of a 4D system that is obtained by coupling of a 1D linear system and a 3D modified generalized Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest are the observations that the hyperchaotic system has a hyperchaotic attractor with three positive Lyapunov exponents under a unique equilibrium, three or infinite equilibria, and there are three types of coexisting attractors of this new 5D hyperchaotic system. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, Poincare projections and power spectrum verifies the existence of the hyperchaotic and chaotic attractors. Moreover, stability of hyperbolic or non-hyperbolic equilibria and two complete mathematical characterization for 5D Hopf bifurcation are rigorously studied. Finally, some electronic circuits are designed to implement the 5D hyperchaotic system.

A New 6D Hyperchaotic System with Four Positive Lyapunov Exponents Coined

This paper reports the finding of a new six-dimensional (6D) autonomous hyperchaotic system, which is obtained by coupling a 1D linear system and a 5D hyperchaotic system that is constructed by adding a linear feedback controller and a nonlinear feedback controller to the Lorenz system. This hyperchaotic system has very simple algebraic structure but can exhibit complex dynamical behaviors. Of particular interest is that it has a hyperchaotic attractor with four positive Lyapunov exponents and a unique equilibrium in a large range of parameters. Numerical analysis of phase trajectories, Lyapunov exponents, bifurcation, power spectrum and Poincaré projections verifies the existence of the hyperchaotic and chaotic attractors. In addition, stability of the hyperbolic equilibrium is analyzed and two complete mathematical characterizations for 6D Hopf bifurcation are given.

Computation of a Canonical Form for Linear 2D Systems

Symbolic computation techniques are used to obtain a canonical form for polynomial matrices arising from discrete 2D linear state-space systems. The canonical form can be regarded as an extension of the companion form often encountered in the theory of 1D linear systems. Using previous results obtained by Boudellioua and Quadrat (2010) on the reduction by equivalence to Smith form, the exact connection between the original polynomial matrix and the reduced canonical form is set out. An example is given to illustrate the computational aspects involved.

Evaluation of Two Acceleration Techniques in a Multithreaded 2D Poisson Equation Solver

Two acceleration techniques, based on additive corrections are evaluated with a multithreaded 2D Poisson equation solver. The popular multigrid algorithm with 2-level grid is compared with the traditional block-correction strategy. In both, single-processor and distributed architectures, block correction is faster than the multigrid due mainly to the smaller cost that the solution of a 1D linear system has over one 2D linear system. Results in both cluster tested show that block correction can reduce significantly the computing time in the solution of very large linear systems. These calculations confirm that the Red/Black ordering is effective only if data fit entirely in cache memory.

Strong practical stability and stabilization of differential linear repetitive processes

Differential linear repetitive processes evolve over a subset of the upper-right quadrant of the 2D plane where the unique feature is a series of sweeps or passes through a set of dynamics governed by the solution of a linear matrix differential equation over a finite duration t ∈ [ 0 , α ] where α is termed the pass length or duration. On each pass an output, termed the pass profile, is produced, which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. The result can be oscillations in the pass-to-pass direction that cannot be controlled by direct application of standard, or 1D linear systems theory. The existing stability theory demands a bounded-input bounded-output property uniformly, which in the case of the along-the-pass dynamics means for t ∈ [ 0 , ∞ ] and for ( k , t ) ∈ [ 0 , ∞ ] × [ 0 , ∞ ] ⊃ [ 0 , ∞ ] × [ 0 , α ] where the integer k ≥ 0 denotes the pass number or index. The pass length is always finite, however, and hence this stability theory could well be too strong in many cases and, in particular, impose very strong conditions in terms of control law design. This paper develops an alternative in such cases by relaxing the requirement for the bounded-input bounded-output property to hold when k → ∞ and t → ∞ simultaneously, provides an explanation of the implications of this in the frequency domain, and then develops control law design algorithms.

Asymptotic stability of positive 2D linear systems with delays

Asymptotic stability of positive 2D linear systems with delays New necessary and sufficient conditions for the asymptotic stability of positive 2D linear systems with delays described by the general model, Fornasini-Marchesini models and Roesser model are established. It is shown that checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to the checking of the asymptotic stability of corresponding positive 1D linear systems without delays. The efficiency of the new criterions is demonstrated on numerical examples.

Error and timing analysis of multiple time-step integration methods for molecular dynamics

Molecular dynamics simulations of biomolecules performed using multiple time-step integration methods are hampered by resonance instabilities. We analyze the properties of a simple 1D linear system integrated with the symplectic reference system propagator MTS (r-RESPA) technique following earlier work by others. A closed form expression for the time step dependent Hamiltonian which corresponds to r-RESPA integration of the model is derived. This permits us to present an analytic formula for the dependence of the integration accuracy on short-range force cutoff range. A detailed analysis of the force decomposition for the standard Ewald summation method is then given as the Ewald method is a good candidate to achieve high scaling on modern massively parallel machines. We test the new analysis on a realistic system, a protein in water. Under Langevin dynamics with a weak friction coefficient ( ζ = 1 ps −1 ) to maintain temperature control and using the SHAKE algorithm to freeze out high frequency vibrations, we show that the 5 fs resonance barrier present when all degrees of freedom are unconstrained is postponed to ≈ 12 fs . An iso-error boundary with respect to the short-range cutoff range and multiple time step size agrees well with the analytical results which are valid due to dominance of the high frequency modes in determining integrator accuracy. Using r-RESPA to treat the long range interactions results in a 6× increase in efficiency for the decomposition described in the text.

Laplacian transfer across a rough interface: numerical resolution in the conformal plane

We use a conformal mapping technique to study the Laplacian transfer across a rough interface. Natural Dirichlet or Von Neumann boundary condition are simply read by the conformal map. Mixed boundary condition, albeit being more complex can be efficiently treated in the conformal plane. We show in particular that an expansion of the potential on a basis of evanescent waves in the conformal plane allows to write a well-conditioned 1D linear system. These general principle are illustrated by numerical results on rough interfaces.