Controlling Chaos
Abstract
Chaos theory is the study of systems that evolve in highly irregular ways due to sensitivity to initial conditions. These include but are not limited to certain physical, biological, chemical, and economic systems. Some theoretical features of chaos are unpredictability and a lack of repetition or pattern. The logistic map is a toy model, commonly used to study the general features of chaos. The logistic map model takes the following form: where depends on the prior term in the sequence . Certain values of reliably yield predictable patterns; we call these stable r-values. Other values of reliably yield chaos; these are chaotic r-values. We developed a method of making a chaotic trajectory more predictable in the short term. We achieved this by alternating between a chaotic value of and a stable value of that are relatively close. In addition to controlling chaos, our project is to forecast and to detect the onset of chaos in otherwise stable systems. We will show how our algorithm can find order embedded and hidden in chaotic trajectories. Our protocol exposes a characteristic dominant frequency for each value of r. We developed the model where the inverse characteristic frequency is an integer multiple k of the cycle n that results from the logistic map model for the corresponding r-value. We will also share how this model can be applied to probe, possibly control, or prevent chaotic behavior in real world systems such as cardiac arrhythmias.
Keywords
chaos, control, logistic map, chaos theory, unstable rhythms
Controlling Chaos
CASB 105
Chaos theory is the study of systems that evolve in highly irregular ways due to sensitivity to initial conditions. These include but are not limited to certain physical, biological, chemical, and economic systems. Some theoretical features of chaos are unpredictability and a lack of repetition or pattern. The logistic map is a toy model, commonly used to study the general features of chaos. The logistic map model takes the following form: where depends on the prior term in the sequence . Certain values of reliably yield predictable patterns; we call these stable r-values. Other values of reliably yield chaos; these are chaotic r-values. We developed a method of making a chaotic trajectory more predictable in the short term. We achieved this by alternating between a chaotic value of and a stable value of that are relatively close. In addition to controlling chaos, our project is to forecast and to detect the onset of chaos in otherwise stable systems. We will show how our algorithm can find order embedded and hidden in chaotic trajectories. Our protocol exposes a characteristic dominant frequency for each value of r. We developed the model where the inverse characteristic frequency is an integer multiple k of the cycle n that results from the logistic map model for the corresponding r-value. We will also share how this model can be applied to probe, possibly control, or prevent chaotic behavior in real world systems such as cardiac arrhythmias.