Lípidos, glicemia, tensión arterial e índice de masa corporal

2.3

Dynamical systems approach

A dynamical system of general form is often expressed by

dx

dt = v(x, t), (2.2)

x(t0)= x0. (2.3)

In these equations, t represents time and it is the independent variable, x(t), represents the state of the system at time t and it is the dependent variable. The vector function v(x, t) typically satisfies some level of continuity. As time evolves, solutions of Eqs. (2.2), (2.3) trace out curves, or in dynamical systems terminology, they flow along their trajectory. Numerical solutions of Eqs. (2.2), (2.3) can almost always be found, however such solutions by themselves are not very desirable for general analysis. While the exact solution of Eqs. (2.2), (2.3) would be ideal, unless v(x, t) is a linear function of the state x and independent of time t, and a few other cases, there is no general way to determine the analytic solution of Eqs. (2.2), (2.3).

When v is independent of time t the system is known as time-independent, or autonomous, and there are some standard techniques for studying these systems. For instance, we can understand the global flow geometry of time-independent systems by studying invariant manifolds of the fixed points of Eqs. (2.2), (2.3), in particular stable and unstable manifolds often play the most important role since they form the skeleton of all possible motion. These concepts are described

in the following. A fixed point of v is a point xf _{such that v(x}f_{) = 0. The}

stable manifolds of a fixed point are all trajectories which reach the fixed point

when t → ∞. In a similar way, the unstable manifolds of the fixed point are

all trajectories which reach the fixed point when t → −∞. The term invariant

manifold is the mathematician notion of material curve. This means that the trajectory of any condition starting on the manifold, must remain on the curve. Often, stable and unstable manifolds act as separatrices, which separate regions of different motion.

Let us consider the planar, frictionless pendulum to explain these concepts. This setup has a point mass, m, at the end of a weightless rod of length l, as shown in Fig. 2.1. The dynamics is given by the second order equation

d2_θ

where g denotes the acceleration due to gravity. Since any nth-order system is

equivalent to a set of first order equations, let us define x= θ and y = ˙θ, which

allows us to write Eq.(2.4) as

˙y= −mgl sin x , ˙x = y, (2.5)

which places it in the first-order vector form given by Eq.(2.2).

Figure 2.1:Pendulum setup

Fig. 2.2 shows the phase portrait of the pendulum. Due to all values for the

positionsθ are consistent for values over the interval from -π to π, we only show

the phase portrait over this interval. If we equalize positions in this way, then θ = π is equivalent to θ = −π, however they appear as two separate points

in the phase portrait, although one can imagine this by putting togetherθ = π

andθ = −π in a cylinder. The fixed points of the pendulum are (θ, ˙θ)=(0, 0)

and (θ, ˙θ)=(π, 0). The fixed point (π, 0) is hyperbolic. In a system of differential equations a stationary hyperbolic point is a point such that the eigenvalues of the linearized system have non-zero real part, and is the point where the stable and unstable manifolds intersect. We will see later that hyperbolicity plays an important role in transport. The stable and unstable manifolds of the fixed point (π, 0) are shown in Fig. 2.2 (blue lines) and form separatrices. These separatrices divide the flow into regions of distinct dynamics: inside these separatrices, the pendulum oscillates (green lines) and outside the separatrices, the pendulum spins in one direction (red lines).

There are many methods for computing the stable and unstable manifolds in time-independent systems. Many of them, are based on growing manifolds of

2.3. DYNAMICAL SYSTEMS APPROACH

Figure 2.2:Pendulum phase portrait

their hyperbolic fixed points. This can be done because in this systems the stable and unstable manifolds of a fixed point are tangent to the eigenvectors of the linearized vector field over such point. However in time-dependent systems (v depends on time t) there are not fixed points (the fixed points change in time), and in addition asymptotic limits for such systems are often meaningless be- cause also they change on time.

Time-dependent dynamical systems typically have regions of dynamically dis- tinct behavior which can be thought of as being divided by separatrices. How- ever, for such systems these regions change over time, and hence so do the separatrices, and one should not read into the analogy between these separatri- ces and traditional definitions of stable and unstable manifolds too much. The concept of hyperbolic point in a time-independent flow can be generalized to time-dependent flows as hyperbolic fluid particle trajectory. It could be thought of as a moving hyperbolic point. Its stable and unstable manifolds are the timedependent generalizations of the separatrices of the saddle type hyperbolic point of the steady flow (Mancho et al., 2004). We refer to theses separatrices as Lagrangian Coherent Structures (LCS), a name which is common in fluid me-

chanics. They are material curves, which means that they cannot be crossed by other fluid particle trajectories.

To find separatrices in time-dependent systems, we use an approach similar to the time-independent systems to try to indirectly obtain the manifolds. We will do this by studing the qualitatively behavior of trajectories near such structures. The indirect method will avoid us the challenging task of having to locate fixed points. This is explained in the following.

Figure 2.3:Two points on either side of a separatrix will diverge from each

other

Fig. 2.3 shows a generic hyperbolic point and its associated stable and unstable manifolds. If we integrate two points initiated on both side of a stable manifold, then these points will diverge from each other forward in time. In a similar way, if we initiated two points on either side of an unstable manifold, then these points will diverge from each other backward in time. This is the reason why these manifolds are called separatrices, since they separate qualitatively different trajectories. Thus we use the viewpoint that, since separatrices divide