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The derivative x′ is the instantaneous rate of change, and F specifies how that rate depends on state and time. An Euler approximation would use x_next=x+dt·F(x,t). The neighboring project family is a particular scalar relaxation; this general ODE notation does not identify its physical meaning.
Terms & definitions
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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
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Original context
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## 3.1 Family 0: Continuous Dynamics
```
f₀(x) = x + 0.1 · (dx_norm - x)
```
**Equation Type**: First-order ODE approximation: x' = F(x,t)
**Explanation**: Models smooth, continuous change toward the input signal. The 0.1 coefficient creates a gradual approach, preventing sudden jumps. This family handles flowing, analog-like processes.
## 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)Exact original notation
First-order ODE approximation: x' = F(x,t)