Attractors are behavioral or structural states toward which, under some specific context, the system converges from a large set of initial states. In some sense these final states attract the neighboring ones. The set of all initial states from which the system’s behavior converges toward the attractor is called a basin of attraction. For example, one can metaphorically consider attractor states to be seas or oceans and their basins of attraction all the places from which rivers flow towards them. The river waters are (gravitationally) attracted towards their attractors (seas and oceans). In a sense, they are destined to attain the final attractor state. Rivers from different continents empty into the Atlantic ocean. All these regions form the basin of attraction of the Atlantic ocean. In similar vein, the ground (minimum) energy levels in atoms are the attractors and all the energy levels from which electrons tend to attain the ground state form the basin of attraction. Atoms may be defined as basins of attraction of their minimum energy levels (attractors). Long term stable shapes of leaves or trees toward which they converge during the growth may be considered as attractors. Remembering (memories) and recognition as well as biological movements are strongly dependent on attractor states in neural networks.
(see Figure 1.)

Figure 1. Potential landscape V (1,2) as a function of state variables 1 and 2. Open circles are repellers,  i.e. unstable, points of the landscape. Any state located at the repeller will tend to leave that point. Black circles located at potential minima are attractor states (attractors). They are stable states of the landscape. Note that between any two stable states (attractors) there is a barrier on the top of which there is the repeller. They attract all the states which belong to their basin of attraction (black lines converging to the attractors). Any state which is in the basin of attraction of a certain attractor point will move towards it and eventually stay at that point. All these states within the basin of attraction including the repellers but excluding the attractor point itself are unstable states. The metaphor of rivers and oceans mentioned in the text may be used here. Rivers would originate at repellers or within the basins of attraction of attractors and tend to leave their origins because they are unstable states. They will flow toward the attractor points because they are the stable states.

On the other hand, repellers are considered unstable states of the system’s behavior. If for some reason the system attains that state (for example by external perturbation) it tends to leave it spontaneously and converge to the attractor state. Hence, between the repeller and the attractor exists a flow or a vector field that acts as an attracting force.  For example, mountain ridges can be considered as repellers. As the mountain snow melts, the waters on each side of these ridges form different rivers which flow toward different seas. They create different basins of attraction. They are watersheds. The highest energy level in atoms may be considered as a repeller because the electron that have attained it can leave the atom and join another one or can converge toward the attractor (the ground energy state) of the first one.  At critical points initial states of matter become repellers (e.g. demagnetized state) and converge toward newly formed attractor (magnetizedstate. At these, so called, phase transition/bifurcation/critical or tipping points, attractor states may destabilize and become repellers and other system states become attractors.

Video: Robert Hristovski, Aleksandar Aceski

Video 1: After an initial perturbation (injecting energy by moving the legs) the exerciser initiates oscillatory dynamics. Due to friction (palm contact with the bar and air resistance) the initial energy injected is being slowly dissipated. Because of energy dissipation the oscillations are dampened and their amplitude θ continuously decreases. The oscillation dynamics of the exerciser is a long transient state toward the attractor of the rest state θ = 0. It is this state towards which the behavior of the oscillation converges. Once the rest state is reached, if the exerciser does not inject additional energy and perturb the rest state, he will remain in it infinitely long.

The initial high amplitude of oscillation as well as all others transient values of amplitude (i.e. θ > 0),  except the θ = 0 (the rest state) are unstable states. They lie onto the basin of attraction of the rest state attractor  (θ = 0). The basin of attraction is the whole curved line along which the ball oscillates.  For any initial amplitude of oscillation that is located on the curved line, the ultimate destiny of the oscillations will be the rest state attractor θ = 0.  This state attracts all other unstable (θ > 0) transient states.

Robert Hristovski 26.03.2015