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Two types of stability of the solution of Equation (6.1.1) are distinguished: stability to the initial condition perturbations for the fixed field v and stability to perturbations of the field v including perturbations of the parameter a. In climate theory it is the second type of stability that is associated with the concept of sensitivity. The term 'robustness' is applied when the same problems are considered in the theory of dynamical systems. The dynamical system is identified as robust (lattice-stable) if small perturbations of the field v do not change its topological properties. The simplest example of the robust system is the system (6.1.1) having only an equilibrium or time-periodic solution whose stability spectrum is located in the left-hand side of the complex plane.

Systems whose limiting states are exhausted by time-periodic and equilibrium regimes have been a principal subject of interest in mathematics and applications. The reasons for such a constraint are well known: for a small dimension of the phase space (n = 1, 2), the field of any smooth system may be transformed by small deformations into the field of the Morse-Smale system, i.e. the system in which all motions at t oo tend to periodic trajectories (including equilibrium states) with a number of time-periodic solutions being finite and no spectra crossing the imaginary axis. However, for n > 3 the situation is changed drastically: there are robust dynamical systems which have different and generally more complicated forms of asymptotes at large t. Furthermore, for large dimensions the robust systems become, in a sense, non-typical.

Direct and adjoint equations of sensitivity. Let us consider three problems associated with Equation (6.1.1): the Cauchy problem, the problem of the construction of equilibrium solutions and the problem of the construction of time-dependent solutions. Each of these problems may be formulated as the problem of the determination of solutions of the functional equation

with a suitable choice of the operator G. An analysis of the sensitivity of solutions of the problems mentioned above is performed by using an identical scheme. We shall describe the essence of this in brief.

There is a broad class of non-linear systems whose response to small perturbations of parameters is almost linear: the world is linear in the small if it is not pathological. According to Arnold, the class of pathological systems forms a 'thin set' within a set of all systems. The class of non-pathological systems is described by the implicit function theorem (Sattinger, 1980).

If conditions of the theorem are fulfilled the determination of the solution for one value a of the parameter allows us to describe system states for all a quite close to a without solving Equation (6.1.2) once again. In the case where G is a scalar function of the scalar variables y, a the theorem guarantees correctness of the formula:

where £ is defined by the relation ldG

Having defined the derivatives in a reasonable way we can extend the relations (6.1.3) and (6.1.4) to the infinite-dimensional functional space.

Let us denote a vector with components y, a by z and assume that the operator G is Frechet differentiable. The operator G is said to be Frechet differentiate at a point z = (y, a) provided there is a bounded linear operator A from a set V to Y such that the quantity R(z, ft) = G(z + ft) — G(z) — Ah is o(h) as || ft || ->• 0, that is, lim hftir^Wz, ft)ii 0.

In the case being investigated,

(a - a), where dG/dy and dG/da are linear operators from V to Y. Certainly, these operators depend on z = (y, a), with the dependence being non-linear.

We now write down the formulation of the implicit function theorem. Let G be a Fréchet differentiable operator. Suppose G(y, â) = 0 and dG(y, a)/dy is an invertible operator. Then for sufficiently small ||a — â|| there is a Fréchet differentiable function y(a) satisfying the relations (6.1.3) and (6.1.4). Furthermore, in a sufficiently small neighbourhood V c V there is the only solution of Equation (6.1.2).

If G has k derivatives the function y(a) has k continuous derivatives, too. Equations for derivatives are found by formal differentiation of the Equation (6.1.2) with respect to a. Basic facts of finite-dimensional analysis (the rule of differentiation of a superposition, Taylor's theorem and the Lagrange formula) have their analogues in the non-linear functional analysis. In particular, as applied to the theorem conditions, the function y(a) can be approximated up to quantities of the order of ||a — <x||* by the appropriate Taylor polynomial. In accordance with (6.1.4), evaluation of the operator

Sensitivity of the climatic system 295

C is reduced to solving the direct equations of sensitivity:

If y is a function of spatial variables and time it is important to know averaged values of this function determined by the functional R(y, a). Let R be a differentiable functional. Its response to perturbations of the parameter a may be computed by the formula

R(y, a) - R(y, a) = ( ^ (y, a) )(y - y) + ( ^ (y, a) )(cc - a)

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