Exponentially stable solution

This is the so-called linear stability condition of the Symplectic Euler method if hQ < 2 the integrator is stable. When hQ > 2, the eigenvalues of the discretization method are both real, with one strictly inside and one strictly outside the unit circle. This implies that the method will exhibit exponentially growing solutions. We say that the stability threshold of the Symplectic Euler method is 2/f2. 

Behavior of Solutions of Nonlinear Systems with Lag in the Vicinity of Exponentially Stable Toroidal Manifolds... 

The locally stable solutions of the TV-charge surface Coulomb problem are constrained solely by spherical boundary conditions and the 0(4) symmetry of the Coulomb interaction. The exponential growth of the multiplicity of solutions—M(N) e Eq. (3.2a)—shows that these restrictions are compatible with a great variety of geometric structures. Only in the simplest systems is there an overlap with the criteria of strict regularity... 

We find that the trivial solution is unstable whereas the periodic solution i.e., the limit cycle) is exponentially stable. 

Since (8.22) has an exponentially stable equilibrium point at the origin, all solutions that start from initial conditions, satisfying r /( + tan l ku 0) + cii 0)) < 0 and m(0) > —Q and do not touch the 77 = 0 line, reach the origin (steady-sliding state) exponentially. If any of these trajectories reach the N = 0 line say at f = q, then the motion stops instantaneously and starts from the rest at u ti), —Q). This pattern continues and may result in a limit cycle at steady state. 

In the ha-hPY sae this describes the inside of a circle centered at point (-1,0) and of radius 1 passing through the origin at (0,0) and the point (-2,0). For a solution with a real decaying exponential, X will have only a real negative part and a stable solution will require that -2/Re(/l) = -2 a rapidly decaying solution with a large value of X this will require a very small time step, h. This... 

V-halTf hpITf This will be satisfied by any value in the negative half plane, i.e. by any value of ha<0. Thus for an exponentially decaying solution, the trapezoidal algorithm will be stable for any desired step size, similar to the backwards difference algorithm. 

Figure 10.2 illustrates the stability regions of the three algorithms in the ha - hp plane. For most engineering problems one deals with differential equations that have stable solutions as time approaches infinity. This means that typically differential equations have a negative real part of the exponential term... 

If Xi and A,2are real numbers and both have negative values, the values of the exponential terms and hence the magnitudes of the perturbations away from the steady-state conditions, c, and T, will reduce to zero, with increasing time. The system response will therefore decay back to its original steady-state value, which is therefore a stable steady-state solution or stable node. 

As discussed in Section 2.5, the solution is therefore physically stable. When t- oo all but the exponential term with the zero eigenvalue tend to zero, possibly after a few oscillations if some eigenvalues are complex. The concentrations are relaxing towards the steady-state given by... 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

When the Hopf bifurcation at p is supercritical (/ 2 < 0) the system has just a single stable limit cycle. This emerges at p and exists across the range p < p < p, within which it surrounds the unstable stationary-state solution. The limit cycle shrinks back to zero amplitude at the lower bifurcation point p%. This behaviour is qualitatively the same as that shown with the simplifying exponential approximation and is illustrated in Fig. 5.4(a). 

The previous two chapters have considered the stationary-state behaviour of reactions in continuous-flow well-stirred reactions. It was seen in chapters 2-5 that stationary states are not always stable. We now address the question of the local stability in a CSTR. For this we return to the isothermal model with cubic autocatalysis. Again we can take the model in two stages (i) systems with no catalyst decay, k2 = 0 and (ii) systems in which the catalyst is not indefinitely stable, so the concentrations of A and B are decoupled. In the former case, it was found from a qualitative analysis of the flow diagram in 6.2.5 that unique states are stable and that when there are multiple solutions they alternate between stable and unstable. In this chapter we become more quantitative and reveal conditions where the simplest exponential decay of perturbations is replaced by more complex time dependences. 

It must be stable. That is, the numerical solutions must not diverge exponentially with time from the true solutions. 

A typical time response for a short-circuited photocurrent in the presence of hydroquinone ( Q) as an added solution redox species is shown in Figure 9. These photocurrents were stable for several hours. In the absence of in the electrolyte, the photocurrent also increased rapidly upon the onset of illumination, but subsequently decayed exponentially to 70% of its initial value in a half-decay time of ca. 25 s. This behavior is similar to that observed for chlorophyll monolayers deposited on SnC (12). Photocurrents under potentially-controlled conditions were also stable upon illumination, but exhibited slower decay characteristics when the light was turned off. This effect is unusual and is currently under further investigation. 

How the perturbations affect the state of the system depends on the eigenvalues Kk. If any eigenvalue has a positive real part, then the solution x grows exponentially, and the corresponding eigenvectors are known as unstable modes. If, on the other hand, all the eigenvalues have negative real parts then a perturbation around the stationary state exponentially decays and the system returns back to its stable state. The linear stability analysis is valid for small perturbations ( x / A v 1) only. 

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