Dynamics
In this chapter, we turn our attention to continuous dynamical systems, which are governed by first and second order linear systems of ordinary differential equations. Such systems, whose unvarying equilibria were the subject of Chapter 6, include the dyn
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Dynamics In this chapter, we turn our attention to continuous dynamical systems, which are governed by first and second order linear systems of ordinary differential equations. Such systems, whose unvarying equilibria were the subject of Chapter 6, include the dynamical motions of mass–spring chains and structures, and the time-varying voltages and currents in simple electrical circuits. Dynamics of continuous media, including fluids, solids, and gases, are modeled by infinite-dimensional dynamical systems described by partial differential equations, [61, 79], and will not be treated in this text, nor will we venture into the vastly more complicated realm of nonlinear dynamics, [34, 41]. Chapter 8 developed the basic mathematical tools — eigenvalues and eigenvectors — used in the analysis of linear systems of ordinary differential equations. For a first order system, the resulting eigensolutions describe the basic modes of exponential growth, decay, or periodic behavior. In particular, the stability properties of an equilibrium solution are (mostly) determined by the eigenvalues. Most of the phenomenology inherent in linear dynamics can already be observed in the two-dimensional situation, and we devote Section 10.3 to a complete description of first order planar linear systems. In Section 10.4, we re-interpret the solution to a first order system in terms of the matrix exponential, which is defined by analogy with the usual scalar exponential function. Matrix exponentials are particularly effective for solving inhomogeneous or forced linear systems, and also appear in applications to geometry, computer graphics and animation, theoretical physics, and mechanics. As a consequence of Newton’s laws of motion, mechanical vibrations are modeled by second order dynamical systems. For stable configurations with no frictional damping, the eigensolutions constitute the system’s normal modes, each periodically vibrating with its associated natural frequency. The full dynamics is obtained by linear superposition of the periodic normal modes, but the resulting solution is, typically, no longer periodic. Such quasi-periodic motion may seem quite chaotic — even though mathematically it is merely a combination of finitely many simple periodic solutions. When subjected to an external periodic forcing, the system usually remains in a quasi-periodic motion that superimposes a periodic response onto its own internal vibrations. However, attempting to force the system at one of its natural frequencies, as prescribed by its eigenvalues, may induce a resonant vibration, of progressively unbounded amplitude, resulting in a catastrophic breakdown of the physical apparatus. In contrast, frictional effects, depending on first order derivatives/velocities, serve to damp out the quasi-periodic vibrations and similarly help mitigate the dangers of resonance.
10.1 Basic Solution Techniques Our initial focus will be on systems du = Au dt
© Springer International Publishing AG, part of Springer Nature 2018 P. J. Olver, C. Shakiban, Applied Linear Algebra, Und
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