Dynamical Systems & Stability

A dynamical system describes the evolution of a state over time according to a deterministic rule. In the continuous case, this is often expressed via autonomous differential equations:

$\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}),\mathbf{x}\in {\mathbb{R}}^n$

where $\mathbf{f}$ is a vector field. In the discrete case, it follows an iterated map:

${\mathbf{x}}_{n+1}=\mathbf{g}({\mathbf{x}}_n)$

1. State Space and Flows

The state space $X$ (often a manifold) contains all possible states of the system. A flow is a mapping $\Phi :\mathbb{R}\times X\to X$ that satisfies the group properties:

1. $\Phi (0,\mathbf{x})=\mathbf{x}$ (Identity)
2. $\Phi (t,\Phi (s,\mathbf{x}))=\Phi (t+s,\mathbf{x})$ (Additivity)

For a system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$, the flow ${\Phi }_t({\mathbf{x}}_0)$ represents the position of a particle at time $t$ that started at ${\mathbf{x}}_0$ at $t=0$.

2. Fixed Points and Orbits

A fixed point (or equilibrium) ${\mathbf{x}}^{\ast }$ is a state where the system remains constant:

• Continuous: $\mathbf{f}({\mathbf{x}}^{\ast })=0$
• Discrete: $\mathbf{g}({\mathbf{x}}^{\ast })={\mathbf{x}}^{\ast }$

The orbit (or trajectory) of ${\mathbf{x}}_0$ is the set $\gamma ({\mathbf{x}}_0)=\{{\Phi }_t({\mathbf{x}}_0):t\in I\}$, where $I$ is the maximal interval of existence.

3. Stability Analysis

Stability characterizes how a system responds to small perturbations from a fixed point ${\mathbf{x}}^{\ast }$.

Lyapunov Stability

${\mathbf{x}}^{\ast }$ is Lyapunov stable if for every $ϵ>0$, there exists $\delta >0$ such that:

$\Vert {\mathbf{x}}_0-{\mathbf{x}}^{\ast }\Vert <\delta ⟹\Vert {\Phi }_t({\mathbf{x}}_0)-{\mathbf{x}}^{\ast }\Vert <ϵ\forall t\ge 0$

Asymptotic Stability

${\mathbf{x}}^{\ast }$ is asymptotically stable if it is Lyapunov stable and there exists $\delta >0$ such that:

$\Vert {\mathbf{x}}_0-{\mathbf{x}}^{\ast }\Vert <\delta ⟹{\lim }_{t\to \infty }{\Phi }_t({\mathbf{x}}_0)={\mathbf{x}}^{\ast }$

The set of all such ${\mathbf{x}}_0$ is the Basin of Attraction.

Lyapunov Functions

A scalar function $V(\mathbf{x})$ is a Lyapunov function for ${\mathbf{x}}^{\ast }$ if $V({\mathbf{x}}^{\ast })=0$, $V(\mathbf{x})>0$ for $\mathbf{x}\ne {\mathbf{x}}^{\ast }$, and $\dot{V}(\mathbf{x})=\nabla V\cdot \mathbf{f}(\mathbf{x})\le 0$. If $\dot{V}<0$, the point is asymptotically stable. This method (Lyapunov’s Direct Method) allows for stability analysis without explicitly solving the differential equations.

4. Linearization and Hartman-Grobman

To analyze a nonlinear system near ${\mathbf{x}}^{\ast }$, we linearize using the Jacobian:

$\dot{\xi }=A\xi ,A=D\mathbf{f}({\mathbf{x}}^{\ast })$

where $\xi =\mathbf{x}-{\mathbf{x}}^{\ast }$.

Hyperbolicity

A fixed point is hyperbolic if all eigenvalues of $A$ have non-zero real parts. This ensures that the qualitative behavior is determined by the linear terms.

Hartman-Grobman Theorem

If ${\mathbf{x}}^{\ast }$ is a hyperbolic fixed point, there exists a homeomorphism $h$ in a neighborhood of ${\mathbf{x}}^{\ast }$ that maps the trajectories of the nonlinear system to those of the linear system $\dot{\xi }=A\xi$. This means the local topology is preserved, allowing us to classify fixed points as sinks, sources, or saddles based on eigenvalues.

5. Invariant Manifolds

For hyperbolic points, the state space decomposes into invariant subspaces based on the eigenvalues of the linearized system:

• Stable Manifold $W^s({\mathbf{x}}^{\ast })$: The set of points that converge to ${\mathbf{x}}^{\ast }$ as $t\to \infty$. It is tangent to the stable subspace $E^s$ formed by eigenvectors associated with eigenvalues $\text{Re}(\lambda )<0$.
• Unstable Manifold $W^u({\mathbf{x}}^{\ast })$: The set of points that converge to ${\mathbf{x}}^{\ast }$ as $t\to -\infty$. It is tangent to the unstable subspace $E^u$ ($\text{Re}(\lambda )>0$).
• Center Manifold $W^c({\mathbf{x}}^{\ast })$: Associated with eigenvalues where $\text{Re}(\lambda )=0$. This manifold is critical because it contains the dynamics that can’t be resolved by linearization alone, such as bifurcations.

6. Bifurcation Theory

A bifurcation is a qualitative change in the topological structure of the flow as a parameter $\mu$ is varied.

• Saddle-Node Bifurcation: Two fixed points (one stable, one unstable) collide and annihilate. Normal form: $\dot{x}=\mu -x^2$.
• Transcritical Bifurcation: Two fixed points exchange stability. Normal form: $\dot{x}=\mu x-x^2$.
• Pitchfork Bifurcation: A fixed point splits into three (or vice versa).
  • Supercritical: $\dot{x}=\mu x-x^3$ (stable 1st equilibrium $\to$ two stable equilibria + one unstable).
  • Subcritical: $\dot{x}=\mu x+x^3$.

7. Discrete Dynamical Systems: The Logistic Map

Discrete systems often exhibit chaotic behavior more readily than continuous ones. The Logistic Map $x_{n+1}=r{x}_n(1-x_n)$ is a foundational model for population dynamics and chaos.

As the growth rate parameter $r$ increases, the system undergoes a period-doubling cascade. At $r\approx 3.5699$, the system enters a chaotic regime where orbits are dense and show sensitive dependence on initial conditions.

import numpy as np
import matplotlib.pyplot as plt

def logistic_map_bifurcation():
    # Number of r values to simulate
    n = 10000
    r = np.linspace(2.5, 4.0, n)
    # Number of iterations total and number of points to plot per r
    iterations = 1000
    last = 100
    # Initialize state
    x = 1e-5 * np.ones(n)

    plt.figure(figsize=(10, 7), dpi=100)
    # Simulate the system
    for i in range(iterations):
        x = r * x * (1 - x)
        # Plot only the last few points to see long-term behavior
        if i >= (iterations - last):
            plt.plot(r, x, ',k', alpha=0.1)

    plt.title("Bifurcation Diagram of the Logistic Map")
    plt.xlabel("Growth Rate (r)")
    plt.ylabel("Population (x)")
    plt.tight_layout()
    plt.show()

# To generate the plot, uncomment the line below:
# logistic_map_bifurcation()

The diagram shows that for $r<3$, there is a single stable fixed point. At $r=3$, it bifurcates into a 2-cycle, then 4-cycle, etc., until chaos is reached.

8. Attractors and Limit Cycles

An attractor is a closed set $A$ to which all nearby orbits converge.

Limit Cycles

A limit cycle is an isolated periodic orbit. In the plane, the Poincaré-Bendixson Theorem provides a powerful tool: if a trajectory is confined to a closed, bounded region containing no fixed points, then it must approach a limit cycle.

A critical implication of this theorem is that chaos cannot occur in ${\mathbb{R}}^1$ or ${\mathbb{R}}^2$ continuous autonomous systems. Continuous chaos requires at least three dimensions (e.g., the Lorenz system in ${\mathbb{R}}^3$).

Chaos and Strange Attractors

In higher dimensions, systems can exhibit Sensitive Dependence on Initial Conditions (the “Butterfly Effect”). Strange attractors, like the Lorenz attractor, have non-integer fractal dimensions and represent complex, non-periodic recurrent behavior in a bounded region of state space.