Suppose u^(0)=1. Find u^(1), u^(10), and u^(20) when (a)...

Suppose $u^{(0)}=1$. Find $u^{(1)}, u^{(10)}$, and $u^{(20)}$ when (a) $u^{(k+1)}=2 u^{(k)}$,
(b) $u^{(k+1)}=-.9 u^{(k)}$,
(c) $u^{(k+1)}=\mathrm{i} u^{(k)}$,
(d) $u^{(k+1)}=(1-2 i) u^{(k)}$.
Is the system stable or unstable? If stable, is it asymptotically stable?

Key Concepts

Complex Numbers in Recurrence Relations

When the multiplier c is a complex number, the stability and behavior of the sequence depend on its modulus. The reasoning remains the same as in the real case: if the modulus is less than one, the sequence tends to zero (asymptotically stable), if it is greater than one, the sequence diverges (unstable), and if it equals one, the sequence may exhibit oscillatory or cyclic behavior without decay, indicating marginal stability. Additionally, the argument (or phase) of the complex number influences the oscillatory dynamics of the sequence.

System Stability in Discrete Dynamical Systems

Stability analysis in discrete dynamical systems examines whether the sequence converges, remains bounded, or diverges over time depending on the multiplier c. For a system defined by u^(k+1) = c · u^(k), the system is asymptotically stable if |c| < 1, meaning that as k increases, u^(k) tends to zero; it is unstable if |c| > 1, resulting in unbounded growth, and if |c| = 1, the system is marginally or neutrally stable, depending on other factors.

First-order Linear Recurrence Relations

A first-order linear recurrence relation defines a sequence where each term is obtained by multiplying the previous term by a constant, i.e., u^(k+1) = c · u^(k). This type of recurrence is fundamental because it leads to solutions that are geometric sequences and serves as a simple model in many applications ranging from finance to population dynamics.

Geometric Sequences

A geometric sequence is one in which each term is obtained by multiplying the previous term by a fixed constant, leading to the general form u^(k) = c^k · u^(0). This concept is central when solving recurrences of the form u^(k+1) = c · u^(k), as it allows for immediately deducing the behavior and closed-form expression for any term in the sequence.

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