Find the (real) stable, unstable, and center subspaces of...

Find the (real) stable, unstable, and center subspaces of the following linear systems: (a) $$ \begin{aligned} & \dot{u}_1=9 u_2 \text {, } \\ & \dot{u}_2=-u_1 \text {; } \\ & \dot{x}_2=3 x_1 \text {; } \\ & \dot{u}_1=u_1-3 u_2+11 u_3, \\ & \end{aligned} $$ (b) $\dot{x}_1=4 x_1+x_2$, (e) $$ \begin{aligned} & \dot{u}_2=2 u_1-6 u_2+16 u_3, \\ & \dot{u}_3=u_1-3 u_2+7 u_3, \end{aligned} \quad \text { (f) } \frac{d \mathbf{u}}{d \ell}=\left(\begin{array}{rrr} -1 & 3 & -3 \\ 2 & 2 & -7 \\ 0 & 3 & -4 \end{array}\right) \mathbf{u} $$ (f) $$ \frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrr} -1 & 3 & -3 \\ 2 & 2 & -7 \\ 0 & 3 & -4 \end{array}\right) \mathbf{u}, $$ (c) $$ \begin{aligned} & \dot{y}_1=y_1-y_2 \\ & \dot{y}_2=2 y_1+3 y_2 \end{aligned} $$ $$ \dot{z}_1=z_2 \text {, } $$ (d) $$ \begin{aligned} & \dot{z}_2=3 z_1+2 z_3, \\ & \dot{y}_3=-z_2 ; \\ & \frac{d \mathbf{u}}{d t}=\left(\begin{array}{llll} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \\ 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \end{array}\right) \mathbf{u} . \end{aligned} $$

Find the (real) stable, unstable, and center subspaces of the following linear systems:
(a)
$$
\begin{aligned}
& \dot{u}_1=9 u_2 \text {, } \\
& \dot{u}_2=-u_1 \text {; } \\
& \dot{x}_2=3 x_1 \text {; } \\
& \dot{u}_1=u_1-3 u_2+11 u_3, \\
&
\end{aligned}
$$
(b) $\dot{x}_1=4 x_1+x_2$,
(e)
$$
\begin{aligned}
& \dot{u}_2=2 u_1-6 u_2+16 u_3, \\
& \dot{u}_3=u_1-3 u_2+7 u_3,
\end{aligned} \quad \text { (f) } \frac{d \mathbf{u}}{d \ell}=\left(\begin{array}{rrr}
-1 & 3 & -3 \\
2 & 2 & -7 \\
0 & 3 & -4
\end{array}\right) \mathbf{u}
$$
(f)
$$
\frac{d \mathbf{u}}{d t}=\left(\begin{array}{rrr}
-1 & 3 & -3 \\
2 & 2 & -7 \\
0 & 3 & -4
\end{array}\right) \mathbf{u},
$$
(c)
$$
\begin{aligned}
& \dot{y}_1=y_1-y_2 \\
& \dot{y}_2=2 y_1+3 y_2
\end{aligned}
$$
$$
\dot{z}_1=z_2 \text {, }
$$
(d)
$$
\begin{aligned}
& \dot{z}_2=3 z_1+2 z_3, \\
& \dot{y}_3=-z_2 ; \\
& \frac{d \mathbf{u}}{d t}=\left(\begin{array}{llll}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 2 \\
1 & 0 & 0 & 0 \\
0 & 2 & 0 & 0
\end{array}\right) \mathbf{u} .
\end{aligned}
$$

Key Concepts

Center Subspace

The center subspace is defined by the eigenvectors corresponding to eigenvalues whose real parts are zero. Dynamics in the center subspace are neutral, meaning that they neither decay nor grow exponentially. Analyzing the center subspace is essential for uncovering bifurcation phenomena and understanding persistent modes that can dictate long-term system behavior.

Linear Dynamical Systems

Linear dynamical systems refer to systems of differential equations where the rate of change of the state vector is given by a linear function of the state. These systems can be represented in matrix form and their behavior is determined by the properties of the system matrix, making them amenable to techniques from linear algebra in stability and phase-space analysis.

Eigenvalue Analysis

Eigenvalue analysis is the process of finding the eigenvalues and eigenvectors of the system matrix. The eigenvalues determine the type of dynamics for each mode of the system: eigenvalues with negative real parts indicate decaying (stable) modes, those with positive real parts indicate growing (unstable) modes, and those with zero real parts indicate modes that neither decay nor grow, often requiring deeper analysis for stability.

Invariant Subspaces

Invariant subspaces are subsets of the state space that are mapped into themselves under the dynamics of the system. In the context of linear systems, these are particularly important because the stable, unstable, and center subspaces are invariant under the system flow, meaning that any initial condition within one of these subspaces will evolve entirely within that subspace.

Stable Subspace

The stable subspace of a linear dynamical system consists of all generalized eigenvectors corresponding to eigenvalues with negative real parts. Trajectories initiated in this subspace decay exponentially toward the equilibrium, and its invariant nature ensures that these modes do not interact with unstable or neutral modes.

Unstable Subspace

The unstable subspace is formed by the collection of eigenvectors associated with eigenvalues with positive real parts. In this subspace, small perturbations grow exponentially over time, leading the system away from equilibrium. Understanding this subspace is critical for controlling or mitigating instability in applications.

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