Fig. 4

Schematic illustration of the evolutionary understanding of the behavioral property. The plot represents the state of the system λ(ɸ) (one arbitrary cycle from − 1.0 to 1.0) over time (horizontal axis). The green line indicates the damping force from model 1 [decay at a maintained value of \( d^{2} \left( {\vec{v}} \right)/dt^{2} \))] between the focal individual and neighbors (or role individual) over time. The contour (black ~ white) represents the 24-h circadian process as expressed by [π/2 = 5:00, π = 12:00, π3/2 = 17:00, and (00:00)] according to the optimized value of the system’s state with arbitrary units of − 1 to 1. The dotted lines show the observations from model 2 (experimental results). The black line denotes the temperature (T) process according to the circadian cycle. The blue line and shade (distribution) show the observed normal states of the biological system according to the circadian temperature cycle. The red line and shade (distribution) denote the observed abnormal states in the perturbed circadian temperature conditions. The dots surrounded by yellow colors (Response) denote plausible evidence for the association of this property. The crucial variable, which can intuitively be set in both dynamics, can simply be considered as the rate of change between the objects arranged according to a high-sensitivity rule. The results above describe a complex behavior with a divided phase (\( \Delta \omega \)) space in which areas of stability are surrounded by confusion. This implies that although their initial states are almost identical (in a comparison of the middle left area of the plot), the response becomes remarkably different with iteration of \( n \) times