For each of the following systems (i) 𝐮 =( 2 -1 3 -2 ) 𝐮, (ii) 𝐮̇=( 1

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For each of the following systems (i) 𝐮 =( 2 -1 ...

For each of the following systems (i) $\mathbf{\mathbf { u }}=\left(\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right) \mathbf{u}$, (ii) $\dot{\mathbf{u}}=\left(\begin{array}{ll}1 & -1 \\ 5 & -3\end{array}\right) \mathbf{u}$, (iii) $\mathbf{\mathbf { u }}=\left(\begin{array}{cc}-3 & 5 / 2 \\ -5 / 2 & 2\end{array}\right) \mathbf{u}:$ (a) Find the general real solution. (b) Using the solution formulas obtained in part (a), plot several trajectories of each system. On your graphs, identify the eigenlines (if relevant), and the direction of increasing $t$ on the trajectories. (c) Write down the type and stability properties of the system.

For each of the following systems
(i) $\mathbf{\mathbf { u }}=\left(\begin{array}{ll}2 & -1 \\ 3 & -2\end{array}\right) \mathbf{u}$,
(ii) $\dot{\mathbf{u}}=\left(\begin{array}{ll}1 & -1 \\ 5 & -3\end{array}\right) \mathbf{u}$,
(iii) $\mathbf{\mathbf { u }}=\left(\begin{array}{cc}-3 & 5 / 2 \\ -5 / 2 & 2\end{array}\right) \mathbf{u}:$
(a) Find the general real solution. (b) Using the solution formulas obtained in part (a), plot several trajectories of each system. On your graphs, identify the eigenlines (if relevant), and the direction of increasing $t$ on the trajectories. (c) Write down the type and stability properties of the system.

Key Concepts

Stability Analysis

Stability analysis involves determining the long-term behavior of a system’s trajectories, particularly whether they converge to, diverge from, or orbit around equilibrium points. This is typically achieved by examining the eigenvalues of the system's matrix. For example, real positive eigenvalues may indicate instability while negative eigenvalues imply stability. The type of equilibrium (node, saddle, or focus) is also classified based on these eigenvalues.

Phase Portraits

Phase portraits provide a graphical representation of the trajectories of a dynamic system in its state space. They are used to visualize the direction and stability of the system’s flow, including invariant lines or eigenlines, which are directions aligned with the eigenvectors. These portraits help in understanding how trajectories evolve over time and how stability is manifested in the system.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are central to the analysis of linear systems since they reveal the directions (eigenvectors) along which the system evolves and the rates (eigenvalues) at which these evolutions occur. They are used to decompose the system into independent modes whose behavior (growth, decay, oscillation) determines the overall dynamics of the system, such as stability and the nature of trajectories.

Linear Dynamical Systems

This concept pertains to the study of systems in which the evolution of variables over time is governed by linear differential equations or linear mappings. The behavior of these systems can be fully captured by matrices, and their evolution can be analyzed using linear algebra techniques. The key idea is to understand how initial states evolve through the repeated application of a linear transformation, often visualized through phase portraits.

General Solution of Linear Systems

The general solution of a linear system is often expressed as a linear combination of modes corresponding to the eigenvalues and eigenvectors of the system's matrix. This formulation uses the principle of superposition inherent in linear systems, allowing the solution to be decomposed into contributions from each eigenvector, and the behavior of the system is clear by examining these individual modes.

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