Lyapunov Exponents: Detailed Explanation, Proofs, and Derivations
Lyapunov exponents are a measure of the rates of separation of infinitesimally close trajectories in dynamical systems. They provide insight into the stability and predictability of a system. A positive Lyapunov exponent indicates chaos, where small differences in initial conditions lead to exponential divergence of trajectories.
Definition of Lyapunov Exponents
Consider a dynamical system given by the differential equation
$$ \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}), $$
where \( \mathbf{x} \in \mathbb{R}^n \) and \( \mathbf{f}: \mathbb{R}^n \to \mathbb{R}^n \). Let \( \mathbf{x}(t) \) be a trajectory of the system with initial condition \( \mathbf{x}(0) = \mathbf{x}_0 \). The Lyapunov exponent \( \lambda \) is defined as
$$ \lambda = \lim_{t \to \infty} \lim_{\| \delta \mathbf{x}(0) \| \to 0} \frac{1}{t} \ln \frac{\| \delta \mathbf{x}(t) \|}{\| \delta \mathbf{x}(0) \|}, $$
where \( \delta \mathbf{x}(t) \) is an infinitesimal perturbation of the trajectory \( \mathbf{x}(t) \).
Properties of Lyapunov Exponents
1. **Invariant Under Coordinate Transformations**: Lyapunov exponents are invariant under smooth coordinate transformations.
2. **Dependence on Initial Conditions**: Lyapunov exponents generally depend on the initial conditions of the system.
3. **Existence of Multiple Exponents**: For an \( n \)-dimensional system, there are \( n \) Lyapunov exponents.
Computation of Lyapunov Exponents
To compute the Lyapunov exponents, consider the Jacobian matrix \( \mathbf{J}(\mathbf{x}) = \frac{\partial \mathbf{f}}{\partial \mathbf{x}} \). The evolution of a perturbation \( \delta \mathbf{x} \) is given by the variational equation
$$ \dot{\delta \mathbf{x}} = \mathbf{J}(\mathbf{x}) \delta \mathbf{x}. $$
Let \( \mathbf{D}(t) = \frac{\partial \mathbf{x}(t)}{\partial \mathbf{x}(0)} \) be the fundamental matrix solution of the variational equation. The Lyapunov exponents are given by the logarithms of the eigenvalues of the matrix
$$ \lim_{t \to \infty} \left( \mathbf{D}(t)^T \mathbf{D}(t) \right)^{\frac{1}{2t}}. $$
Proof of Invariance Under Smooth Coordinate Transformations
Let \( \mathbf{y} = \mathbf{h}(\mathbf{x}) \) be a smooth coordinate transformation. The dynamical system in the new coordinates is
$$ \dot{\mathbf{y}} = \mathbf{g}(\mathbf{y}) = \mathbf{J}(\mathbf{h}(\mathbf{x})) \mathbf{f}(\mathbf{x}). $$
Let \( \mathbf{D}_\mathbf{y}(t) = \frac{\partial \mathbf{y}(t)}{\partial \mathbf{y}(0)} \). The perturbation in the new coordinates evolves as
$$ \dot{\delta \mathbf{y}} = \mathbf{J}(\mathbf{y}) \delta \mathbf{y}. $$
The Jacobian matrices in the new and original coordinates are related by
$$ \mathbf{J}_\mathbf{y}(\mathbf{y}) = \mathbf{J}(\mathbf{x}) \mathbf{H}(\mathbf{x}), $$
where \( \mathbf{H}(\mathbf{x}) = \frac{\partial \mathbf{h}}{\partial \mathbf{x}} \). Therefore,
$$ \dot{\delta \mathbf{y}} = \mathbf{H}(\mathbf{x}) \mathbf{J}(\mathbf{x}) \delta \mathbf{x}. $$
Since the Lyapunov exponents are determined by the eigenvalues of the fundamental matrix, which are invariant under coordinate transformations, the Lyapunov exponents remain unchanged.
Example: Lyapunov Exponents for the Lorenz System
The Lorenz system is a well-known example of a chaotic system. The equations are given by
$$ \dot{x} = \sigma (y – x), $$
$$ \dot{y} = x (\rho – z) – y, $$
$$ \dot{z} = x y – \beta z, $$
where \( \sigma, \rho, \beta \) are parameters. The Jacobian matrix for the Lorenz system is
$$ \mathbf{J} = \begin{pmatrix} -\sigma & \sigma & 0 \\ \rho – z & -1 & -x \\ y & x & -\beta \end{pmatrix}. $$
The Lyapunov exponents can be computed numerically by integrating the variational equations along with the Lorenz equations and applying the method described above.
