# Chaotic dynamics and diffusion in a piecewise linear equation

## Abstract

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed usingmore »

- Authors:

- Department of Mathematics, Shah Jalal University of Science and Technology, Sylhet–3114 (Bangladesh)
- Department of Physiology, 3655 Promenade Sir William Osler, McGill University, Montreal, Quebec H3G 1Y6 (Canada)
- Department of Mathematics and Statistics, University of Victoria, P.O. Box 1700 STN CSC, Victoria, British Columbia V8W 2Y2 (Canada)

- Publication Date:

- OSTI Identifier:
- 22402535

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 25; Journal Issue: 3; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; DIFFERENTIAL EQUATIONS; DIFFUSION; DYNAMICS; EIGENVALUES; FEEDBACK; FOUR-DIMENSIONAL CALCULATIONS; GENES; HALL EFFECT; INTERACTIONS; NONLINEAR PROBLEMS; OSCILLATIONS; PERIODICITY; TRAJECTORIES

### Citation Formats

```
Shahrear, Pabel, E-mail: pabelshahrear@yahoo.com, Glass, Leon, E-mail: glass@cnd.mcgill.ca, and Edwards, Rod, E-mail: edwards@uvic.ca.
```*Chaotic dynamics and diffusion in a piecewise linear equation*. United States: N. p., 2015.
Web. doi:10.1063/1.4913417.

```
Shahrear, Pabel, E-mail: pabelshahrear@yahoo.com, Glass, Leon, E-mail: glass@cnd.mcgill.ca, & Edwards, Rod, E-mail: edwards@uvic.ca.
```*Chaotic dynamics and diffusion in a piecewise linear equation*. United States. doi:10.1063/1.4913417.

```
Shahrear, Pabel, E-mail: pabelshahrear@yahoo.com, Glass, Leon, E-mail: glass@cnd.mcgill.ca, and Edwards, Rod, E-mail: edwards@uvic.ca. Sun .
"Chaotic dynamics and diffusion in a piecewise linear equation". United States.
doi:10.1063/1.4913417.
```

```
@article{osti_22402535,
```

title = {Chaotic dynamics and diffusion in a piecewise linear equation},

author = {Shahrear, Pabel, E-mail: pabelshahrear@yahoo.com and Glass, Leon, E-mail: glass@cnd.mcgill.ca and Edwards, Rod, E-mail: edwards@uvic.ca},

abstractNote = {Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.},

doi = {10.1063/1.4913417},

journal = {Chaos (Woodbury, N. Y.)},

number = 3,

volume = 25,

place = {United States},

year = {Sun Mar 15 00:00:00 EDT 2015},

month = {Sun Mar 15 00:00:00 EDT 2015}

}