Radialrecurrent

Radialrecurrent is a concept used to describe a class of dynamical models in which the evolution of the system depends primarily on the radial distance from a fixed center and exhibits recurrence, meaning the process tends to revisit regions of the state space over time. The central idea is to emphasize radial structure and temporal return behavior rather than full dependence on exact position in space.

In formal terms, let c be a fixed center in the state space and define r_t as the distance from the current state to c, r_t = d(S_t, c). A radialrecurrent model updates its state based largely on r_t, potentially together with a limited history. Deterministic formulations may define S_{t+1} = f(S_t, r_t), while stochastic variants use a transition mechanism that depends predominantly on r_t. Across formulations, the angular coordinates around c are either neglected or treated as secondary, leading to rotational symmetry about the center.

Key properties of radialrecurrent systems include a strong emphasis on radial symmetry, a dependence on radial distance rather than full position, and a recurrence characteristic where trajectories repeatedly approach previously visited radii or radial bands. Variants include radialrecurrent graphs, which organize nodes in concentric layers around a hub, and radialrecurrent neural structures that combine radial basis functions with recurrent processing.

Applications of radialrecurrent concepts arise in modeling phenomena with central attractors or hubs, such as diffusion with symmetry, navigation tasks around a central target, and data-processing architectures that exploit radial locality. See also: radial basis function, recurrent neural network, Markov chain, dynamical systems.