At all instants, elements, objects, events, and properties have variable values. Variable-value sets are system states {state, system} {system state}. Systems can be in stable, unstable, or cyclic equilibrium or in stable steady state.
Systems have state sequences {trajectory, system}|. Trajectories are paths through phase space.
Systems can go to final state, repeat state, or never repeat, using constantly interfering states. System can have interactions that make states that interact to form temporary substates {progression}.
Tables {state transition graph} show all possible state transitions.
System behaviors, once begun, can have no further regulation and so follow trajectory {ballistic trajectory}|.
States can have trajectories from all other states {ergodic process}|. Trajectories can have equal or unequal probabilities. Ergodic processes always have non-recurring loops, because system eventually returns to previous states. Because probabilistic, ergodic processes do not have recurrence.
If they have no attractors, have no loops, are deterministic, and have conservation, systems can run in reverse {reversible trajectory}.
Over time, systems tend to move to terminal state {attractor}|, such as constant flow. Trajectories near state tend toward state {attraction basin} {basin of attraction}. Attractors that have more trajectories going to them have higher probability.
Trajectories can result in state {catastrophe, system}| that stops trajectory or changes available states.
Trajectories can go from one state to another, with no trend toward final state {chaos, system}| [Gleick, 1987] [Lorenz, 1963].
Ordered non-equilibrium systems {dissipative system} can have steady state, because matter and/or energy flow through.
Stable systems {stability, system} have attractors with short state-cycle length and large attraction basins, so changing from one state to nearby states, or changing transition rule slightly, leaves system in same attraction basin. In unstable systems, if state or transition rule slightly changes, system changes to long state-cycle length {chaos} or changes attraction basins {catastrophe}. If network is sparsely connected, network tends to stay in same attraction basin. If network is densely connected, network tends to be chaotic or catastrophic.
In deterministic complex systems, trajectories tend to go to repeated states {state cycle}| {recurrence} {oscillator, system}. Examples are pendulums, fluids, circuits, and lasers.
length
Cycles have number of steps {length, cycle}. If nodes have input from all other nodes, length is square root of number of states and is large, and number of attractors is number of nodes divided by natural-logarithm base e and is small. If nodes have two inputs, length is square root of number of nodes and is small, and number of attractors is number of nodes divided by natural-logarithm base e and is small.
3-Computer Science-System Analysis
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Date Modified: 2022.0225