# Controlling chaos faster

## Abstract

Predictive feedback control is an easy-to-implement method to stabilize unknown unstable periodic orbits in chaotic dynamical systems. Predictive feedback control is severely limited because asymptotic convergence speed decreases with stronger instabilities which in turn are typical for larger target periods, rendering it harder to effectively stabilize periodic orbits of large period. Here, we study stalled chaos control, where the application of control is stalled to make use of the chaotic, uncontrolled dynamics, and introduce an adaptation paradigm to overcome this limitation and speed up convergence. This modified control scheme is not only capable of stabilizing more periodic orbits than the original predictive feedback control but also speeds up convergence for typical chaotic maps, as illustrated in both theory and application. The proposed adaptation scheme provides a way to tune parameters online, yielding a broadly applicable, fast chaos control that converges reliably, even for periodic orbits of large period.

- Authors:

- Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen (Germany)
- (BCCN), 37077 Göttingen (Germany)
- (Germany)

- Publication Date:

- OSTI Identifier:
- 22351013

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 24; Journal Issue: 3; Other Information: (c) 2014 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; ASYMPTOTIC SOLUTIONS; CHAOS THEORY; CONTROL; CONVERGENCE; FEEDBACK; INSTABILITY; ORBITS; PERIODICITY; VELOCITY

### Citation Formats

```
Bick, Christian, Bernstein Center for Computational Neuroscience, Institute for Mathematics, Georg–August–Universität Göttingen, 37073 Göttingen, Kolodziejski, Christoph, III. Physical Institute—Biophysics, Georg–August–Universität Göttingen, 37077 Göttingen, Timme, Marc, and Institute for Nonlinear Dynamics, Georg–August–Universität Göttingen, 37077 Göttingen.
```*Controlling chaos faster*. United States: N. p., 2014.
Web. doi:10.1063/1.4895848.

```
Bick, Christian, Bernstein Center for Computational Neuroscience, Institute for Mathematics, Georg–August–Universität Göttingen, 37073 Göttingen, Kolodziejski, Christoph, III. Physical Institute—Biophysics, Georg–August–Universität Göttingen, 37077 Göttingen, Timme, Marc, & Institute for Nonlinear Dynamics, Georg–August–Universität Göttingen, 37077 Göttingen.
```*Controlling chaos faster*. United States. doi:10.1063/1.4895848.

```
Bick, Christian, Bernstein Center for Computational Neuroscience, Institute for Mathematics, Georg–August–Universität Göttingen, 37073 Göttingen, Kolodziejski, Christoph, III. Physical Institute—Biophysics, Georg–August–Universität Göttingen, 37077 Göttingen, Timme, Marc, and Institute for Nonlinear Dynamics, Georg–August–Universität Göttingen, 37077 Göttingen. Mon .
"Controlling chaos faster". United States.
doi:10.1063/1.4895848.
```

```
@article{osti_22351013,
```

title = {Controlling chaos faster},

author = {Bick, Christian and Bernstein Center for Computational Neuroscience and Institute for Mathematics, Georg–August–Universität Göttingen, 37073 Göttingen and Kolodziejski, Christoph and III. Physical Institute—Biophysics, Georg–August–Universität Göttingen, 37077 Göttingen and Timme, Marc and Institute for Nonlinear Dynamics, Georg–August–Universität Göttingen, 37077 Göttingen},

abstractNote = {Predictive feedback control is an easy-to-implement method to stabilize unknown unstable periodic orbits in chaotic dynamical systems. Predictive feedback control is severely limited because asymptotic convergence speed decreases with stronger instabilities which in turn are typical for larger target periods, rendering it harder to effectively stabilize periodic orbits of large period. Here, we study stalled chaos control, where the application of control is stalled to make use of the chaotic, uncontrolled dynamics, and introduce an adaptation paradigm to overcome this limitation and speed up convergence. This modified control scheme is not only capable of stabilizing more periodic orbits than the original predictive feedback control but also speeds up convergence for typical chaotic maps, as illustrated in both theory and application. The proposed adaptation scheme provides a way to tune parameters online, yielding a broadly applicable, fast chaos control that converges reliably, even for periodic orbits of large period.},

doi = {10.1063/1.4895848},

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

number = 3,

volume = 24,

place = {United States},

year = {Mon Sep 01 00:00:00 EDT 2014},

month = {Mon Sep 01 00:00:00 EDT 2014}

}