Mathematical and Computational Science: September 2009

Monday, September 28, 2009

Linear dynamical systems

Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).

Flows

For a flow, the vector field Φ(x) is a linear function of the position in the phase space, that is,

$\phi(x) = A x + b\,,$

with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity). The case b ≠ 0 with A = 0 is just a straight line in the direction of b:

$\Phi^t(x_1) = x_1 + b t \,.$

When b is zero and A ≠ 0 the origin is an equilibrium (or singular) point of the flow, that is, if x₀ = 0, then the orbit remains there. For other initial conditions, the equation of motion is given by the exponential of a matrix: for an initial point x₀,

$\Phi^t(x_0) = e^{t A} x_0 \,.$

When b = 0, the eigenvalues of A determine the structure of the phase space. From the eigenvalues and the eigenvectors of A it is possible to determine if an initial point will converge or diverge to the equilibrium point at the origin.
The distance between two different initial conditions in the case A ≠ 0 will change exponentially in most cases, either converging exponentially fast towards a point, or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence. For nonlinear systems this is one of the (necessary but not sufficient) conditions for chaotic behavior.

Geometrical definition

A dynamical system is the tuple $\langle \mathcal{M}, f , \mathcal{T}\rangle$ , with $\mathcal{M}$ a manifold (locally a Banach space or Euclidean space), $\mathcal{T}$ the domain for time (non-negative reals, the integers, ...) and f an evolution rule t→f^t (with $t\in\mathcal{T}$ ) such that f^t is a diffeomorphism of the manifold to itself. So, f is a mapping of the time-domain $\mathcal{T}$ into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain $\mathcal{T}$ .

Measure theoretical definition

A dynamical system may be defined formally, as a measure-preserving transformation of a sigma-algebra, the quadruplet (X,Σ,μ,τ). Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X,Σ) is a measurable space. μ is a finite measure on the sigma-algebra, so that the triplet (X,Σ,μ) is a probability space. A map $\tau:X\to X$ is said to be Σ-measurable if and only if, for every $\sigma \in \Sigma$ , one has $\tau^{-1}\sigma \in \Sigma$ . A map τ is said to preserve the measure if and only if, for every $\sigma \in \Sigma$ , one has μ(τ ^{− 1}σ) = μ(σ). Combining the above, a map τ is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The quadruple (X,Σ,μ,τ), for such a τ, is then defined to be a dynamical system.
The map τ embodies the time evolution of the dynamical system. Thus, for discrete dynamical systems the iterates $\tau^n=\tau \circ \tau \circ \ldots\circ\tau$ for integer n are studied. For continuous dynamical systems, the map τ is understood to be finite time evolution map and the construction is more complicated.

Thursday, September 24, 2009

Quaternions and spatial rotation

Unit quaternions provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more numerically stable and may be more efficient. Quaternions have found their way into applications in computer graphics, robotics, navigation, molecular dynamics and orbital mechanics of satellites.

The complex numbers can be defined by introducing an abstract symbol i which satisfies the usual rules of algebra and additionally the rule i² = −1. This is sufficient to reproduce all of the rules of complex number arithmetic: for example, $(a+b\mathbf{i})(c+d\mathbf{i}) = ac + ad\mathbf{i} + b\mathbf{i}c + b\mathbf{i}d\mathbf{i} = ac + ad\mathbf{i} + bc\mathbf{i} + bd\mathbf{i}^2 = (ac - bd) + (bc + ad) \mathbf{i}$ .
In the same way the quaternions can be defined by introducing abstract symbols i, j, k which satisfy the rules i² = j² = k² = ijk = −1 and the usual algebraic rules except the commutative law of multiplication (a familiar example of such a noncommutative multiplication is matrix multiplication). From this all of the rules of quaternion arithmetic follow: for example, one can show that $(a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}) (e + f\mathbf{i} + g\mathbf{j} + h\mathbf{k}) = (ae - bf - cg - dh) + (af + be + ch - dg) \mathbf{i} + (ag + ce + df - bh) \mathbf{j} + (ah + de + bg - cf) \mathbf{k}$ .
The imaginary part $b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$ of a quaternion behaves like a vector $\vec{v} = (b,c,d)$ in three dimension vector space, and the real part a behaves like a scalar in $\mathbb{R}$ . When quaternions are used in geometry, it is more convenient to define them as a scalar plus a vector: $a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} = a + \vec{v}$ .
Those who have studied vectors at school might find strange to add a number to a vector, as they are objects of very different natures, or to multiply two vectors together, as this operation is usually undefined. However, if one remembers that it is a mere notation for the real and imaginary parts of a quaternion, it becomes more legitimate.
We can express quaternion multiplication in the modern language of vector cross and dot products (which were actually inspired by the quaternions in the first place). In place of the rules i² = j² = k² = ijk = −1 we have the vector multiplication rule $\vec{v} \vec{w} = \vec{v} \times \vec{w} - \vec{v} \cdot \vec{w}$ , where:

$\vec{v} \vec{w}$ is vector multiplication,
$\vec{v} \times \vec{w}$ is vector cross product (a vector),
$\vec{v} \cdot \vec{w}$ is vector scalar product (a number).

Vector multiplication is noncommutative (because of the cross product), while scalar-scalar and scalar-vector multiplications commute. From these rules it follows immediately that $(s + \vec{v}) (t + \vec{w}) = (s t - \vec{v} \cdot \vec{w}) + (s \vec{w} + t \vec{v} + \vec{v} \times \vec{w})$ .
The (left and right) multiplicative inverse of a nonzero quaternion is $(s + \vec{v})^{-1} = \frac{s - \vec{v}}{s^2 + |\vec{v}|^2}$ , as can be verified by direct calculation.

Mathematical and Computational Science