# Kinematics in Biology: Symbolic Dynamics Approach

## Abstract

**:**

## 1. Introduction

## 2. Kinematics of Isolated Animal

- (1)
- Metabolism or internal dynamics as the only cause of motion: Isolated animal.
- (2)
- Motion is composed of discrete linear steps.
- (3)
- There is a maximal displacement, in one step, which is a consequence of the finite size of the animal and the finite energy at disposal.
- (4)
- Any step depends on the previous step taken: Markov hypothesis.

- (5)
- Differentiability/Continuity: close steps give origin to close steps.
- (6)
- Isotropy: there is no preferential direction for the motion.
- (7)
- One parameter characterizing the iterated process so that there exist for some values of the parameter, at least, one chaotic attractor.

- (a)
- From the assumption (1) the movement of an animal in isolation depends on its internal state or metabolism. This does not mean that the motion is simple. Moreover, the internal state of the animal may change with time and with the energy spent in the motion. Therefore, next to these basic assumptions we have to consider, theoretically and empirically, the dependence of patterns of motion according to the behavior associated with the internal states of the animal. This issue will be considered in future work in detail. Here, we simple aim to describe geometrically short time patches of trajectories, seen as templates or archetype movements, in which the animal behavior is ideally stable. An implicit assumption here is that we have a correspondence between animal behavior and the parameters determining the geometry trajectory.
- (b)
- If eventually, the empirical data shows that the trajectories are not well described by linear steps, then the assumption (2) can be read as the trajectories following approximately the linear step, that is, are contained in a rectangle of $\delta $ wide which contains the linear step. This point is discussed in Section 5.
- (c)
- There is no special reason for the assumption (5). The alternative would need eventually a cause or an explanation for the existing discontinuity or point without derivative. We could assume that there are discontinuity points (in a finite number) for some reason. Eventually, empirical data may require this adjustment. In future work we plan to deal with this issue.
- (d)
- The existence of a chaotic attractor assumed in (7) allows the simulation of the animal movement with a deterministic model. The variability of the trajectories is codified in the initial condition for the chaotic system and in its parameters. In a certain sense, due to the chaotic nature of the model we can see it as a pseudo random number generator which is adequate to produce typical trajectories with varied geometrical and statistical properties. We can chose initial conditions randomly or if we pretend a particular trajectory, with prescribed characteristics, we can obtain it, since the process is deterministic.
- (e)
- Different types of coordinates can be considered, using the above assumptions. We have tested polar coordinates and coupled Cartesian coordinates. However, in the present paper, as in Reference [10], we focus on Cartesian uncoupled coordinates. A general framework using matricial iteration is very promising and will be used in forthcoming paper. There are several reasons to focus, on a first approach, on the very simple iterated maps with uncoupled Cartesian coordinates: (1) its symbolic dynamics is well known, (2) the two or three dimensional trajectories can be reduced to one dimensional dynamics (3) the complexity of trajectories obtained is sufficiently interesting and varied to be analyzed.
- (f)
- Since we are considering isolated animal if there is symmetry breaking this means that the animal has in its internal processes a preference to some direction. For example, the possibility of detecting magnetic field, light, heat or other phenomena. This point is very interesting for certain classes of animals, however in this case we do not have isolation.

## 3. Bimodal Maps

#### 3.1. Symbolic Dynamics for Bimodal Maps

#### 3.2. Topological Markov Chain

**Example**

**1.**

## 4. Isentropic Motion and Unique Attractor Cases

#### 4.1. Piecewise Linear Semi-Conjugation and Isentropics

#### 4.2. Unique Critical Orbit

**Lemma**

**1.**

**Proof.**

**Example**

**2.**

**Example**

**3.**

## 5. The Model—Cartesian Plane Motion

#### 5.1. Definition

#### 5.2. Interpretation of the Model

#### 5.3. Pure and Mixed Bimodal Trajectories

#### 5.4. Scale, Uncertainty and Precision in the Trajectory Determinacy

**Example**

**4.**

## 6. Dictionary

## 7. Conclusions, Discussion and Future Work

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Graphs of ${\tau}_{-1,s,r}$ for $s=2$, and $r=0,-0.05,-0.1,-0.15,-0.2$, on the left and ${\tau}_{1,s,r}$ for $s=2,$$r=0,0.05,0.1,0.15,0.2$, on the right. The growth number is equal to 2 in every case, and, therefore, give origin to the same topological entropy $log2.$ Note that ${\tau}_{\rho ,s,r}\left(-1\right)={\tau}_{\rho ,s,r}\left(1\right)$, in every case.

**Figure 3.**The region $\Omega $ with the isentropes represented as black curves: for ${h}_{t}=log2$, the two largest curves connecting the points $L,E$ and $M,F$, and for ${h}_{t}=log\left(1+\sqrt{3}\right)$ the smallest curves near the points G and K.

**Figure 4.**The graphics of ${f}_{b,d}$ with the trajectories of the critical points, ${c}_{-}$ and ${c}_{+}$ in the case $b=0.9862072184965917,d=0$, the kneading invariant is $\left(RR{M}^{\infty},LL{M}^{\infty}\right).$ The growth number is $s=1+\sqrt{3}$ and $r=0.$

**Figure 5.**The graphics of ${f}_{b,d}$ with the trajectories of the critical points, ${c}_{-}$ and ${c}_{+}$ in the case $b=0.984713955503324,d=-0.00924702872344659.$ the kneading invariant is $\left(RR{\left({C}_{+}LL\right)}^{\infty},LL{\left({C}_{+}LL\right)}^{\infty}\right).$ The growth number is $s=1+\sqrt{3}$ and $r=-3s\left(s+1\right)$.

**Figure 6.**For parameters $b=-0.97$ and $d=0$, the initial conditions ${s}_{x,0}=0.8718539083,{s}_{y,0}=0.7322656727$ and ${\tilde{s}}_{x,0}={s}_{x,0}+{10}^{-11},{\tilde{s}}_{y,0}={s}_{y,0}+{10}^{-11}$ lead to different trajectories after few time steps. In this case approximately 20 time steps.

**Figure 7.**Two patches of pure bimodal trajectories with the same parameters $b=-0.877,d=0$, which compose the mixed bimodal trajectory on the right, with a rotation of $20\xb0$ between them.

**Figure 8.**Two patches of pure bimodal trajectories with the different parameters $b=-0.97,d=0$, on the left, $b=-0.877,d=0$, on the right, which compose the mixed bimodal trajectory on the right, with a rotation of $20\xb0$ between them.

**Figure 9.**Trajectory region obtained by the iteration of the map ${F}_{b,d}$, with $\delta =0.075$ and $\tilde{\delta}=0.1$, for the dimensions of the rectangular regions and disks respectively. On the left an example of an eventual real trajectory and in the middle the trajectory fitted into the pure bimodal trajectory region.

**Figure 10.**Trajectory obtained with ${s}_{x,0}=0.2$, ${s}_{y,0}=-0.54$, $b=-0.92$, $d=0.002$, $\delta =0.075$ and $\tilde{\delta}=0.1$, for the dimensions of the rectangular regions and disks respectively. The real trajectory is contained inside the region determined by the sequence of rectangles and disks. Therefore, the characteristic lengths $\delta $ and $\tilde{\delta}$ give an estimate of the precision involved in the measurements.

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Correia Ramos, C.
Kinematics in Biology: Symbolic Dynamics Approach. *Mathematics* **2020**, *8*, 339.
https://doi.org/10.3390/math8030339

**AMA Style**

Correia Ramos C.
Kinematics in Biology: Symbolic Dynamics Approach. *Mathematics*. 2020; 8(3):339.
https://doi.org/10.3390/math8030339

**Chicago/Turabian Style**

Correia Ramos, Carlos.
2020. "Kinematics in Biology: Symbolic Dynamics Approach" *Mathematics* 8, no. 3: 339.
https://doi.org/10.3390/math8030339