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F maps a current state and input to the next state. This is the general form of a driven discrete dynamical system. Reproduction requires the actual mapping, initial state, and input sequence; the nearby weighted-average family supplies one particular example.
Terms & definitions
The recovered passage does not supply a separate symbol table. Read the source context for the notation used in this version; familiar letters may carry different meanings in other builds.
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
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## 3.2 Family 1: Discrete Dynamics
```
f₁(x) = 0.9 · x + 0.1 · utility
```
**Equation Type**: Discrete state transition: x_{t+1} = F(x_t, u_t)
**Explanation**: Implements exponential smoothing between current state and utility signal. The 90/10 split creates memory of past states while responding to new inputs. Used for step-by-step reasoning.
## 3.3 Family 2: Reaction-Diffusion
```
f₂(x) = x + 0.05 · (softsign(dx_norm) - x)
```
**Equation Type**: Simplified reaction-diffusion: ∂x/∂t = D∇²x + f(x)Exact original notation
Discrete state transition: x_{t+1} = F(x_t, u_t)