Singular perturbation

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In mathematics, a perturbed problem has a singular perturbation if its solution $φ(x)$ cannot be uniformly approximated by an asymptotic expansion

$\varphi(x) \approx \sum_{n=0}^N \delta_n(\epsilon) \psi_n(x) \,$

as $\epsilon \to 0$ . Here $δ n (ε)$ are a sequence of functions of $ε$ of increasing order, such as $δ n (ε) = ε n$ . Singularly perturbed systems are generally characterized by dynamics operating on multiple scales.

1 Examples in time
2 Examples in space
3 An algebraic example
4 Methods of analysis
5 References

[edit] Examples in time

An electrically driven robot manipulator can have slower mechanical dynamics and faster electrical dynamics, thus exhibiting two time scales. In such cases, we can divide the system into two subsystems, one corresponding to faster dynamics and other corresponding to slower dynamics, and then design controllers for each one of them separately. Through a singular perturbation technique, we can make these two subsystems independent of each other, thereby simplifying the control problem.

Consider a class of system described by following set of equations:

$\dot{x}_1 = f_1(x_1,x_2) + \epsilon g_1(x_1,x_2,\epsilon), \,$

$\epsilon\dot{x}_2 = f_2(x_1,x_2) + \epsilon g_2(x_1,x_2,\epsilon), \,$

$x_1(0) = a_1, x_2(0) = a_2, \,$

with $0<\epsilon <\!\!< 1$ . The second equation indicates that the dynamics of $x 2$ is much faster than that of $x 1$ . A certain mathematical theorem states that, with the correct conditions on the system, it will initially and very quickly approximate the solution to the equations

$\dot{x}_1 = f_1(x_1,x_2), \,$

$f_2(x_1,x_2) = 0, \,$

$x_1(0)=a_1,\,$

on some interval of time and that, as $ε$ decreases toward zero, the system will approach the solution more closely in that same interval.^[1]

[edit] Examples in space

In fluid mechanics, the properties of a slightly viscous fluid are dramatically different outside and inside a narrow boundary layer. Thus the fluid exhibits multiple spatial scales.

Reaction-diffusion systems in which one reagent diffuses much more slowly than another can form spatial patterns marked by areas where a reagent exists, and areas where it does not, with sharp transitions between them. In ecology, predator-prey models such as

$u_t = \epsilon u_{xx} + uf(u) - vg(u), \,$

$v_t = v_{xx} + vh(u), \,$

where $u$ is the prey and $v$ is the predator, have been shown to exhibit such patterns. ^[2]

[edit] An algebraic example

Consider the problem of finding all roots of the polynomial $ε x 3 - x 2 + 1$ . In the limit $\epsilon\to 0$ , this cubic degenerates into the quadratic $1 - x 2$ with roots at $x = \pm 1$ . Singular perturbation analysis suggests that the cubic has another root $x \approx 1/\epsilon\,$ . Indeed, with $ε = 0.1$ , the roots are -0.955, 1.057, and 9.898. With $ε = 0.01$ , the roots are -0.995, 1.005, and 99.990. With $ε = 0.001$ , the roots are -0.9995, 1.0005, and 999.999.

Here again, in a sense, the problem has multiple scales. Two of the roots converge to finite numbers as $ε$ decreases, while the third becomes arbitrarily large.

[edit] Methods of analysis

A perturbed problem whose solution can be approximated on the whole problem domain, whether space or time, by a single asymptotic expansion has a regular perturbation. Most often in applications, an acceptable approximation to a regularly perturbed problem is found by simply replacing the small parameter $ε$ by zero everywhere in the problem statement. This corresponds to taking only the first term of the expansion, yielding an approximation that converges, perhaps slowly, to the true solution as $ε$ decreases. The solution to a singularly perturbed problem cannot be approximated in this way. As seen in the examples above, a singular perturbation generally occurs when a problem's small parameter multiplies its highest operator. Thus naively taking the parameter to be zero changes the very nature of the problem. In the case of differential equations, boundary conditions cannot be satisfied; in algebraic equations, the possible number of solutions is decreased.

Singular perturbation theory is a rich and ongoing area of exploration for mathematicians, physicists, and other researchers. The methods used to tackle problems in this field are many. The more basic of these include the method of matched asymptotic expansions and WKB approximation for spatial problems, and in time, the Poincaré-Lindstedt method and periodic averaging.

[edit] References

^ Ferdinand Verhulst. Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Springer, 2005. ISBN 0-387-22966-3.
^ M. R. Owen and M. A. Lewis. How Predation can Slow, Stop, or Reverse a Prey Invasion, Bulletin of Mathematical Biology (2001) 63, 655-684.

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Category: Differential equations