Text 3. Dynamics and Stability
Stability is a dynamical concept. To explain it, we shall use some fundamental notions from the theory of dynamical systems (see, for example, Hirsch and Smale  and Guckenheimer and Holmes ). The laws of dynamics are usually presented as equations of motion, which we write in the abstract form of a dynamical system:
= X(u). (1.7.1)
Here, u is a variable describing the state of the system under study, X is a system-specific function of u, and = du/dt, where t is time. The set of all allowed u's forms the state, or phase space P. We usually view X as a vector field on P. For a classical mechanical system, u is often a 2n-tuple (q1, ... , qn, p1, ... , pn) of positions and momenta, and for fluids, u is a velocity field in physical space.
As time evolves, the state of the system changes; the state follows a curve u(t) in P. The trajectory u(t) is assumed to be uniquely determined if its initial condition u0 = u(0)is specified. An equilibrium stateis a state ue such that X(ue) = 0. The unique trajectory starting at ue is ue itself; that is, ue does not move in time.
The language of dynamics has been an extraordinarily useful tool in the physical and biological sciences, especially during the last few decades. The study of systems that develop spontaneous oscillations through a mechanism called the Poincaré–Andronov–Hopf bifurcation is an example of such a tool (see Marsden and McCracken , Carr , and Chow and Hale , for example). More recently, the concept of "chaotic dynamics" has sparked a resurgence of interest in dynamical systems. This occurs when dynamical systems possess trajectories that are so complex that they behave as if they were, in some sense, random. Some believe that the theory of turbulence will use such notions in its future development. We are not concerned with chaos directly, although it plays a role in some of what follows. In particular, we remark that in the definition of stability below, stability does not preclude chaos. In other words, the trajectories near a stable point can still be temporally very complex; stability just prevents them from moving very far from equilibrium.
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