Abstract
Motion in biology is studied through a descriptive geometrical method. We consider a deterministic discrete dynamical system used to simulate and classify a variety of types of movements which can be seen as templates and building blocks of more complex trajectories. The dynamical system is determined by the iteration of a bimodal interval map dependent on two parameters, up to scaling, generalizing a previous work. The characterization of the trajectories uses the classifying tools from symbolic dynamics—kneading sequences, topological entropy and growth number. We consider also the isentropic trajectories, trajectories with constant topological entropy, which are related with the possible existence of a constant drift. We introduce the concepts of pure and mixed bimodal trajectories which give much more flexibility to the model, maintaining it simple. We discuss several procedures that may allow the use of the model to characterize empirical data.
Highlights
Motion was one of the first topics considered in science
With the enormous variety of phenomena and concepts, it is easy to be overwhelmed with information and interrelated data
We expect that using this method to describe the motion of an animal it is possible the development of a more complete model, maintaining its simplicity, including interactions with the environment and with other animals. This can be pursued allowing the dependence of the parameters on the position or on the displacement, obtaining an larger dynamical system or a non-autonomous discrete dynamical system
Summary
Motion was one of the first topics considered in science. it was precisely from the study of motion that was developed modern science. It is possible to determine, at least, a class of kneading sequences which turns the observed sequence into admissible with respect to the iterated map In this case, the approximate values for the topological entropy and other invariants can be computed, obtaining a characterization of the system which produces similar trajectories, as the given one. We expect that using this method to describe the motion of an animal it is possible the development of a more complete model, maintaining its simplicity, including interactions with the environment and with other animals This can be pursued allowing the dependence of the parameters on the position or on the displacement, obtaining an larger dynamical system or a non-autonomous discrete dynamical system.
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