(a) Show that if 𝐮(t) solves 𝐮=A 𝐮, then 𝐯(t)=𝐮(2 t) solves 𝐯̇=B 𝐯, where B=2 A. (b) How are the

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(a) Show that if 𝐮(t) solves 𝐮=A 𝐮, then 𝐯(t)=𝐮(2 t) solves...

(a) Show that if $\mathbf{u}(t)$ solves $\mathbf{u}=A \mathbf{u}$, then $\mathbf{v}(t)=\mathbf{u}(2 t)$ solves $\dot{\mathbf{v}}=B \mathbf{v}$, where $B=2 A$.
(b) How are the solution trajectories of the two systems related?

Key Concepts

Linear Systems of Differential Equations

A linear system of differential equations, typically written in the form u' = Au, involves a constant matrix A that governs the dynamics of the system. These systems are fundamental in understanding how a set of state variables evolve over time, and their behavior is directly dictated by the properties of the matrix A, such as its eigenvalues and eigenvectors.

Time Reparameterization

Time reparameterization involves changing the variable representing time, for example through a substitution like v(t) = u(s(t)) where s(t) is a new time scale. This concept is useful to compare systems that evolve at different rates and shows that altering the time variable often results in a scaled or modified system, even though the underlying geometric behavior of the trajectories remains unchanged.

Chain Rule in Differentiation

The chain rule is a fundamental calculus principle for differentiating composite functions. In the context of differential equations, when a time substitution is made, the derivative of the outer function must be multiplied by the derivative of the inner time function, introducing a scaling factor in the resulting differential equation.

Phase Portrait and Trajectory Equivalence

The phase portrait of a dynamical system represents the geometric paths (or trajectories) that the state variables follow in the system's state space. When two systems are related by a time scaling, their trajectories are geometrically identical but are traversed at different speeds. This means that while the shape of the solution curves is preserved, the timing along these curves is scaled.

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