Tensions in Practice: Stronger transient transfer in tension with smaller excursions¶
Discrete linear updates · two coupled states
An input state y halves on every tick. A response x also halves, but receives an added contribution from the previous y. A coupling of 2 transfers more of that input into x than a coupling of ½. Both systems have the same long-run decay eigenvalues, yet x rises to 2 in one and only ½ in the other. Eventual decay does not specify the size of the intervening excursion.
Transfer more of a brief input into the response
Use stronger coupling to make the response register a transient input.
Keep the response excursion smaller
Limit how much one state can temporarily drive another.
Why these aims pull against each other
The coupling changes near-term amplification without changing the two eigenvalues. A long-run stability verdict therefore cannot choose the coupling for a peak-sensitive application.
Choose an arrangement to see what changes and what remains difficult.
Finite illustrative comparisons. Text states carry the meaning; color is not a measured score or universal preference.
What this choice protects
What it costs
When it fits
Compare the arrangements
Stronger coupling
At each tick set new x = old x/2 + 2×old y, and new y = old y/2.
| Response x | Input state y | |
|---|---|---|
| Tick 0 | 0 | 1 |
| Tick 1 | 2 | ½ |
| Tick 2 | 2 | ¼ |
| Tick 3 | 1½ | ⅛ |
| Long run | 0 | 0 |
- What it protects
- The decaying input drives a larger response: x reaches 2 at ticks 1 and 2.
- What it costs
- The same transfer produces a larger temporary excursion even though both states eventually approach zero.
- When it fits
- A stronger transient response is useful and its excursion is acceptable.
Illustration note: The states start at x = 0, y = 1. The update is fixed, linear and simultaneous, using only old values. This editorial model has eigenvalues ½ and ½; no nonlinear operating limit is implied.
Weaker coupling
Keep both halving terms but set new x = old x/2 + old y/2.
| Response x | Input state y | |
|---|---|---|
| Tick 0 | 0 | 1 |
| Tick 1 | ½ | ½ |
| Tick 2 | ½ | ¼ |
| Tick 3 | ⅜ | ⅛ |
| Long run | 0 | 0 |
- What it protects
- The maximum response in this trace is ½, four times smaller than with coupling 2.
- What it costs
- The response to the same brief input is also four times weaker; reduced excursion sacrifices the desired transfer if that response matters.
- When it fits
- Peak restraint matters more than response strength, with the weaker signal still useful.
Illustration note: For either coupling c, y at tick n is 2^(−n), and x is c×n×2^(1−n) for n ≥ 1. These formulas establish the long-run limit and the displayed peak; this is not an empirical design recommendation.
What this illustration does—and does not—establish
Eigenvalue and Eigenvector Eigenvalue And Eigenvector: Eigenvalues versus Singular Values (scopal) supplies the distinction between asymptotic decay and transient growth. The two coupling choices are explicit editorial linear systems.
- The comparison holds self-decay, initial state and timing fixed; only coupling changes.
- The response is a state component, not a computed singular value or a universal disturbance norm.
- The same eigenvalues do not mean identical eigenvectors, transient behavior or robustness.
Source entries
Eigenvalue And Eigenvector
Eigenvalue And Eigenvector: Eigenvalues versus Singular Values (scopal) supplies the conflict examined here.
Eigenvalues versus Singular Values (scopal)
The failure mode is using eigenvalues to reason about sensitivity or amplification in a non-symmetric system — concluding a transient cannot grow because all eigenvalues are below one, when transient growth is governed by singular values and can be large despite stable eigenvalues.