Hidden order: How adaptation builds complexity by John Holland

By John Holland

Al readers how Complexity--the watershed technology schedule for no less than the nexttwo decades--is affecting our lives.

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N) denotes the columns of a matrix A; then Im A = span{a1 , a2 , . . , an }. 9) A square matrix U ∈ Rn×n whose columns form an orthonormal basis for Rn is called an orthogonal matrix, and it satisfies U T U = UU T = I . Now let A = [ai j ] ∈ Rn×n ; then the trace of A is defined as n trace (A) := aii . 2 Eigenvalues and Eigenvectors Let A ∈ Rn×n ; then the eigenvalues of A are the n roots of its characteristic polynomial p(λ) = det(λI − A). The spectrum of A is the set of all λ that are eigenvalues of A.

94) for all x0 ∈ Rd . 94) is called the sample Lyapunov exponents of the solution. We therefore see that the trivial solution is almost surely exponentially stable if and only if the sample Lyapunov exponents are negative. The almost sure exponential stability means that almost all sample paths of the solution will tend to the equilibrium position x = 0 exponentially fast. To establish the theorems on the almost sure exponential stability, we need to prepare a useful lemma. We assume that the existence-and-uniqueness is fulfilled and f (0, t) ≡ 0, g(0, t) ≡ 0.

However, many dynamical systems cannot be properly modeled by an ordinary differential equation. In particular, for some particular classes of systems, the future evolution of the state variables x(t) not only depends on their current value x(t0 ), but also on their past values, say x(ξ), t0 − r ≤ ξ ≤ t0 , r > 0. Such systems are called timedelay systems that may arise for a variety of reasons in many scientific disciplines including engineering, biology, ecology, and economics. 1 Functional Differential Equations We can use functional differential equations to describe time-delay systems.

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