Hurwitz determinants

The Routh-Hurwitz conditions are well known and can be used to determine, in principle, the stability properties of the steady state of any n-variable system. This advantage is, however, balanced by the fact that in practice their use is very cumbersome, even for n as small as 3 or 4. The evaluation, by hand, of all the coefficients Cl of the characteristic polynomial and the Hurwitz determinants A constitutes a rather arduous task. It is for this reason that in the past this tool of linear stability analysis could hardly be found in the literature of nonlinear dynamics. The situation changed with the advent of computer-algebra systems or symbolic computation software. Software such as Mathematica (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario) makes it easy to obtain exact, analytical expressions for the coefficients C/ of the characteristic polynomial (1.12) and the Hurwitz determinants A . 

According to (1.38), a Hopf bifurcation occurs if the Hurwitz determinant A3 vanishes, A3 = 0. 

Since T is negative according to the stability conditions (10.23), Ci is always positive. The positivity of the Hurwitz determinant A2 and C4 at the Hopf bifurcation imply that C3 > 0 there and that well defined [205]. Gathering terms of equal powers in k, we rewrite the Hopf condition as... 

The coefficients C K) of the characteristic polynomial det[J( )—X tl4] = 0 and the Hurwitz determinants A K) are easily obtained using computational algebra software such as Mathematic A (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario). The th mode undergoes a stationary bifurcation when condition (12.41d) is violated, namely c K) = 0, as discussed in Sect. 1.2.3, see (1.36). In other words, a Turing bifurcation of the uniform steady state corresponds to c iki) = 0 with k 0, while the stability conditions (12.41) are satisfied for all other modes with k fej. The feth mode undergoes an oscillatory bifurcation when condition (12.41c) is violated, namely A K) = 0, as discussed in Sect. 1.2.3, see (1.38). A wave bifurcation of the uniform steady state corresponds to A k i) = 0 with k f/ 0, while the stability conditions (12.41) are satisfied for all otha modes with k few As discussed in Sect. 10.1.2, see (10.29), a wave bifurcation cannot occur in a two-variable reaction-diffusion system. 

Use the Routh-Hurwitz eriterion to determine the number of roots with positive real parts in the following eharaeteristie equations... 

In the same way that the Routh-Hurwitz criterion offers a simple method of determining the stability of continuous systems, the Jury (1958) stability test is employed in a similar manner to assess the stability of discrete systems. 

If one characteristic exponent has a positive real part, it is unstable. The signs of ht are determined by means of the Hurwitz theorem. 

Recently various computational hydro codes have been adapted to the determination of underwater shock parameters. A Lagrangian code (with artificial viscosity) augmented by a sharp shock routine was used by Sternberg Hurwitz (Ref 12) to generate the curves shown in Figs 22,23 24... 

It is often difficult to determine quickly the roots of the characteristic equation. Hurwitz(I,) and Routh( 0) developed an algebraic procedure for finding the number of roots with positive real parts and consequently whether the system is unstable or not. 

By using the Routh-Hurwitz criterion determine whether or not the system is stable. 

In order to determine the number of roots of the z-transformed characteristic equation that lie outside the unit circle, a procedure analogous to the Routh-Hurwitz approach for continuous systems (Section 7.10.2) can be used. The Routh-Hurwitz criterion cannot be applied directly to the characteristic equation f(z) = 0. However, by mapping the interior of the unit circle in the z-piane on to the left half of a new complex variable -plane, the Routh-Hurwitz criterion can be applied as for continuous systems to the corresponding characteristic equation in terms of the new variable<4,). This mapping can be achieved using the bilinear transformation07 ... 

Hurwitz, A. R. and S. T. Liu. 1977. Determination of aqueous solubility cWjhpalues of estrogend. Pharm. 

For stability at a rest point one wishes to show that the eigenvalues of the linearization lie in the left half of the complex plane. There is a totally general result, the Routh-Hurwitz criterion, that can determine this. It is an algorithm for determining the signs of the real parts of the zeros of a polynomial. Since the eigenvalues of a matrix A are the roots of a polynomial... 

Examination of the characteristic equation indicates that it is not necessary to compute the actual values of the roots. All that is required is a knowledge of the location of the roots, i.e., if the roots lie on the right- or left-hand side of the imaginary axis. A simple test known as the Routh-Hurwitz test allows one to determine if any root is located on the right side of the imaginary axis, therefore rendering the system unstable. 

To determine acceptable values, employ the Routh-Hurwitz approach ... 

The stability of the steady state is determined using the Routh-Hurwitz criterion which states that the Eigenvalues of the Jacobian have all negative real parts when... 

Hurwitz AR and Liu ST, Determination of aqueous solubility and pJCa values of estrogens, /. Pharm. Sci, 66, 62 627 (1977). NB This paper reported otherwise good work that was compromised by poor temperature and ionic strength control. 

To determine the stability of the th mode, we conduct a Routh-Hurwitz analysis. All eigenvalues k have a negative real part, if... 

Generally, inhomogeneities in parameters of an array of reactors lead to nonuniform steady states. This is not the case for Lengyel-Epstein networks with inhomogeneities in the parameter a, as is clear from the structure of (13.139). The network still has a unique uniform steady state given by (13.52). We use the Routh-Hurwitz criterion to determine the stability boundaries of this USS. Note that the Routh-Hurwitz analysis is general and can deal with the case where inhomogeneities in parameters lead to nonuniform steady states. Let... 

Cote P, van der Velde G, Cassidy JD, Carroll LJ, Hogg-Johnson S, Holm LW, Carragee EJ, Haldeman S, Nordin M, Hurwitz EL et al (2008) The burden and determinants of neck pain in workers— results of the bone and joint decade 2000-2010 task force on neck pain and its associated disorders. Eur Spine J 17 S60-S74... 

Reactor oscillator method. The real power of this general method of analysis, however, stems from the fact that it may be inverted in order to obtain the quantity H(joi) experimentally, and thus in principle to determine the function K t). The method was first suggested in 1954 by Hurwitz and Brooks as a method for investigating the stability of the first Experimental Breeder Reactor. If we solve equation (12) for H(s) we find ... 

Hurwitz, Contribution a la Determination des Parametres Cinetiques Reels des Processus d Electrode, Thesis, Brussels, 1964. 

H. D. Hurwitz Professor Chismadjev did not limit himself to the problem of membrane breakdown and membrane fusion which is also tackled by Professor Berg s contribution. Other important aspects of bioelectrochemistry have been treated by Professor Chimadjev, like, for example, the determination of membrane potential, also considered in Professor Adams contribution, the electric field across membrane, the influence of antibiotics and of other vectors for the ion transport. I wish to ask the audience if there are any comments which concern such aspects of the selectivity of cell membranes to ions and the role of peptides. 

In the considered model, electrostatic interaction between protons and metal surface charge determines the distributions of protons and electrostatic potential in the pore. These phenomena distinguish the present pore model from the gas- and electrolyte-filled single pore models pioneered by Srinivasan et al. (1967), Srinivasan and Hurwitz (1967) and De Levie (Levie, 1967). With the explicit consideration of the pore wall surface charge, the potential of zero charge of the catalyst material... 

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