Time dependence of the attainability regions of third order systems

SUMMARY The set of differential equations governing the motion of an unrestrained coned wheelset travelling on a tangent section of track and acted upon by creep forces arising from the contact between wheel and rail are, in the terminology of numerical analysis, extremely "stiff". This stiffness can be attributed to the existence of two negative real eigenvalues in the solution of the eigenproblem associated with the linearized equations of motion. Compared with the two complex conjugate eigenvalues that complete this solution, the real eigenvalues have large magnitudes and necessitate that relatively. small timesteps be used in order to obtain an accurate numerical integration of the full set of equations of motion. However, by truncating the set of left and right eigenvectors to eliminate these real eigenvalues in a modal analysis of the wheelset, it was found that their contribution to the overall dynamic response is negligible. This same modal truncation approach was then applied to the sub-structured equations of motion for a simple rail vehicle system consisting of two wheelsets connected to a main body by linear springs and dampers. Essentially, the physical degrees of freedom for each wheelset substructure were replaced by a single complex coordinate obtained from the previous normal modes analysis. Using this model reduction procedure, accurate numerical results for the motion of the rail vehicle were generated several times faster than the results obtained by numerically integrating the full set of differential equations directly.

In a previous work, we have derived the general solution of the state space linear fractional system of commensurate order for real simple and multiple eigenvalues of the state space matrix. The obtained solutions of the homogeneous and non-homogeneous cases have been expressed as a linear combination of introduced fundamental functions. In this paper, the above work has been extended to solve the state space linear fractional system of commensurate order for complex eigenvalues of the state space matrix. First, suitable fundamental functions corresponding to the different types of complex eigenvalues of the state space matrix are introduced. Then, the derived formulations of the resolution approach are presented for the homogeneous and the non-homogeneous cases. The solutions are expressed in terms of a linear combination of the proposed fundamental functions which are in the form of exponentials, sine, cosine, damped sine and damped cosine functions depending on the commensurate fractional order. The results are validated by solving an illustrative example to demonstrate the effectiveness of the proposed analytical tool for the solution of the state space linear fractional system of commensurate order.

A method for obtaining analytic bounds for period doubling and cyclic fold instability regions in linear time-periodic systems with piecewise constant coefficients and time delay is suggested. The method is based on the use of transition matrices for Meissner’s equation corresponding to the desired type of instability. Analytic expressions for the disconnected regions of fold and flip instability for two- and three-segment coefficients including both complex and real eigenvalues in Meissner’s equation are obtained. The proposed method when applied to the example of two-segment interrupted turning with complex eigenvalues in each segment yields the same results as those obtained recently for the boundaries of the flip regions (Szalai and Stepan, 2006, “Lobes and Lenses in the Stability Chart of Interrupted Turning,” J Comput. Nonlinear Dyn., 1, pp. 205–211). Next, the period-doubling instability regions for a particular delay differential equation related to the damped Meissner’s equation and the fold instabilities for a model of delayed position feedback control are analytically obtained. Finally, we extend the method to a single degree-of-freedom milling model with a three-piecewise-constant-segment approximation to the true specific cutting force in which lower bounds for and horizontal locations of the regions of flip instability are obtained. The analytic results are verified through numerical stability charts obtained using the temporal finite element method. Conditions for the existence of islands of instability are also obtained.

In this chapter a unified approach to the design of observers has been detailed. The algorithm developed is applicable to all categories of observers studied in the literature. The key concept of eigenstructure assignment developed in Chapter 2 is used in the construction of the observer. The observer eigenvalues define the observer stability matrix F, and the eigenvectors define the trans formation matrix T. Defining the stability matrix in the modal canonical form facilitates formulation of a computationally simple observer design algorithm. Further this modal canonical observer structure also enjoys advantages in real time implementation since (i) the stability matrix F has a minimum parameter representation and (ii) only a bank of first-order (real eigenvalues) or second-order (complex conjugate eigenvalues) filters needs to be implemented. Existence conditions of the observer with arbitrary eigenvalues are derived using the Kronecker canonical structure properties of the resulting singular rectangular matrix pencils. The computationally stable SCF is used to extract these structural properties. However, construction of the SCF is only required to check the existence of the solution for a specified observer problem and is not needed for design optimisation of the observer.

We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate eigenvalues, the real ones show characteristics of an intermediate incomplete spectrum, that is, of a so-called thinned ensemble. On the other hand, the complex ones show repulsion compatible with cubic-order repulsion of non-normal matrices for the real matrices, but higher order repulsion for the complex and quaternion matrices.

Eigenvalues and eigenfunctions of the Kramers equation. Application to the Brownian motion of a pendulum

This paper introduces a rational function approximation of the fractional order transfer function \( H(s) = \frac{{(\tau_{0} s)^{\alpha } }}{{[1 + (\tau_{0} s)^{2\alpha } ]}},\quad for\,\,0\, < \,\alpha \, \le \,0.5 \). This fractional order transfer function is one of the fundamental functions of the linear fractional system of commensurate order corresponding to pure complex conjugate poles or eigenvalues, in sα. Hence, the proposed approximation will be used in the solution of the linear fractional systems of commensurate order. Illustrative examples are given to show the exactitude and the efficiency of the approximation method.

We consider linear steady-state systems of third order with one controlling (perturbing) action. The control action is assumed to be limited in absolute value and to be contained in the set of piecewise continuous functions. The case is investigated when our system has two complex conjugated and one real eigenvalues. Variation of the attainability region as time increases is studied. We show analytically that the changing of the attainability region with time variation depends on the correlation between the real parts of the eigenvalues.

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

Real nonsymmetric matrices may have either real or complex conjugate eigenvalues. These matrices can be seen to be pseudosymmetric as ηMη^{-1}=M^{t}, where the metric η could be secular (a constant matrix) or depending upon the matrix elements of M. Here we construct ensembles of a large number N of pseudosymmetric n×n (n large) matrices using N[n(n+1)/2≤N≤n^{2}] independent and identically distributed random numbers as their elements. Based on our numerical calculations, we conjecture that for these ensembles the nearest level spacing distributions [NLSDs, p(s)] are sub-Wigner as p_{abc}(s)=ase^{-bs^{c}}(0<c<2) and the distributions of their eigenvalues fit well to D(ε)=A[tanh{(ε+B)/C}-tanh{(ε-B)/C}] (exceptions also discussed). These sub-Wigner NLSDs are encountered in Anderson metal-insulator transition and topological transitions in a Josephson junction. Interestingly, p(s) for c=1 is called semi-Poisson, and we show that it lies close to the form p(s)=0.59sK_{0}(0.45s^{2}) derived for the case of 2×2 pseudosymmetric matrix where the eigenvalues are most aptly conditionally real, E_{1,2}=a±sqrt[b^{2}-c^{2}], which represent characteristic coalescing of eigenvalues in parity-time (PT) -symmetric systems.

Implications of complex eigenvalues in homogeneous flow: A three-dimensional kinematic analysis

Li-Yorke chaos in weak topology of the n-dimensional linear systems

This paper is describing solutions for singularly perturbed linear systems which are considered in a particularly critical case. The matrix of a linear system has complex conjugate eigenvalues. The eigenvalues ​​of the matrix system under consideration do not have zeros on the boundary of the region under consideration and outside of this region. Imaginary parts of the eigenvalues ​​of the matrix are positive with the exception of boundary points in the considered domain. For evaluation of functions, a proved lemma was used. A uniform approximation was constructed for the solution of the initial Cauchy problem in particularly critical case with a certain degree of accuracy.

As a model to provide a hands-on, elementary understanding of ‘vortex dynamics’, we introduce a piecewise linear non-invertible map called a twisted baker map. We show that the set of hyperbolic repelling periodic points with complex conjugate eigenvalues and that without complex conjugate eigenvalues are simultaneously dense in the phase space. We also show that these two sets equidistribute with respect to the normalised Lebesgue measure, in spite of a non-uniformity in their Lyapunov exponents.

Equivalent higher order linear and non-linear systems