 By Marji Lines Alfredo Medio

A sophisticated undergraduate and graduate textbook at the thought of nonlinear dynamical platforms.

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Additional resources for Non-linear Dynamics: A Primer

Example text

The speed of the motion along the eigenvector uk is given by x(t) ˙ = λk eλk t x(0) = λk x(t), that is, the speed is a constant proportion of the position of x. 19) is invariant. ) If we choose the initial conditions on S so that the coefficients ci of the general solution are all zero except the two corresponding to the pair of ¯ j ), the orbits of the system remain on the plane complex eigenvalues (λj , λ S forever. Again, assuming that the matrix A has m distinct eigenvalues, we divide the eigenvectors (or, in the complex case, the vectors equal to the real and imaginary parts of them) into three groups, according to whether the corresponding eigenvalues have negative, positive or zero real parts.

Exercises 19 of orbits starting from certain practically relevant sets of initial conditions. These properties can often be investigated effectively by a combination of mathematical, geometrical, statistical and numerical methods. Much of what follows is dedicated precisely to the study of some of those methods. Before turning to this goal, however, we review in chapter 2 the class of dynamical systems which is best understood: linear systems. Dynamical linear systems in both continuous and discrete time are not terribly interesting per se because their behaviour is morphologically rather limited and they cannot be used effectively to represent cyclical or complex dynamics.

14) where we have employed Euler’s formula e±iθ = cos θ ± i sin θ to write e(α±iβ)t = eαt [cos(βt) ± i sin(βt)] . 14) can be further simplified by transforming the real and ¯j ) into the imaginary parts of each element of the complex vectors (uj , u 2 If z = α + iβ is an arbitrary complex number, then the complex number z¯ = α − iβ is the complex conjugate of z. The same notation is used for vectors. 28 Review of linear systems polar coordinates (l) (l) (l) (l) (l) (l) aj = Cj cos(φj ) bj = Cj sin(φj ) (l) l = 1, 2, .