### Abstract

Positive Lyapunov exponents measure the asymptotic exponential divergence of nearby trajectories of a dynamical system. Not only they quantify how chaotic a dynamical system is, but since their sum is an upper bound for the rate of information production, they also provide a convenient way to quantify the complexity of a dynamical network. We conjecture based on numerical evidences that for a large class of dynamical networks composed by equal nodes, the sum of the positive Lyapunov exponents is bounded by the sum of all the positive Lyapunov exponents of both the synchronization manifold and its transversal directions, the last quantity being in principle easier to compute than the latter. As applications of our conjecture we: (i) show that a dynamical network composed of equal nodes and whose nodes are fully linearly connected produces more information than similar networks but whose nodes are connected with any other possible connecting topology; (ii) show how one can calculate upper bounds for the information production of realistic networks whose nodes have parameter mismatches, randomly chosen: (iii) discuss how to predict the behavior of a large dynamical network by knowing the information provided by a system composed of only two coupled nodes. (C) 2011 Elsevier B.V. All rights reserved.

Original language | English |
---|---|

Pages (from-to) | 1309-1318 |

Number of pages | 10 |

Journal | Physics Letters A |

Volume | 375 |

Issue number | 10 |

Early online date | 3 Feb 2011 |

DOIs | |

Publication status | Published - 7 Mar 2011 |

### Keywords

- metric invariant
- systems
- automorphisms

### Cite this

*Physics Letters A*,

*375*(10), 1309-1318. https://doi.org/10.1016/j.physleta.2011.01.054

**How complex a dynamical network can be?** / Baptista, M. S.; Kakmeni, F. Moukam; Del Magno, Gianluigi; Hussein, M. S.

Research output: Contribution to journal › Article

*Physics Letters A*, vol. 375, no. 10, pp. 1309-1318. https://doi.org/10.1016/j.physleta.2011.01.054

}

TY - JOUR

T1 - How complex a dynamical network can be?

AU - Baptista, M. S.

AU - Kakmeni, F. Moukam

AU - Del Magno, Gianluigi

AU - Hussein, M. S.

PY - 2011/3/7

Y1 - 2011/3/7

N2 - Positive Lyapunov exponents measure the asymptotic exponential divergence of nearby trajectories of a dynamical system. Not only they quantify how chaotic a dynamical system is, but since their sum is an upper bound for the rate of information production, they also provide a convenient way to quantify the complexity of a dynamical network. We conjecture based on numerical evidences that for a large class of dynamical networks composed by equal nodes, the sum of the positive Lyapunov exponents is bounded by the sum of all the positive Lyapunov exponents of both the synchronization manifold and its transversal directions, the last quantity being in principle easier to compute than the latter. As applications of our conjecture we: (i) show that a dynamical network composed of equal nodes and whose nodes are fully linearly connected produces more information than similar networks but whose nodes are connected with any other possible connecting topology; (ii) show how one can calculate upper bounds for the information production of realistic networks whose nodes have parameter mismatches, randomly chosen: (iii) discuss how to predict the behavior of a large dynamical network by knowing the information provided by a system composed of only two coupled nodes. (C) 2011 Elsevier B.V. All rights reserved.

AB - Positive Lyapunov exponents measure the asymptotic exponential divergence of nearby trajectories of a dynamical system. Not only they quantify how chaotic a dynamical system is, but since their sum is an upper bound for the rate of information production, they also provide a convenient way to quantify the complexity of a dynamical network. We conjecture based on numerical evidences that for a large class of dynamical networks composed by equal nodes, the sum of the positive Lyapunov exponents is bounded by the sum of all the positive Lyapunov exponents of both the synchronization manifold and its transversal directions, the last quantity being in principle easier to compute than the latter. As applications of our conjecture we: (i) show that a dynamical network composed of equal nodes and whose nodes are fully linearly connected produces more information than similar networks but whose nodes are connected with any other possible connecting topology; (ii) show how one can calculate upper bounds for the information production of realistic networks whose nodes have parameter mismatches, randomly chosen: (iii) discuss how to predict the behavior of a large dynamical network by knowing the information provided by a system composed of only two coupled nodes. (C) 2011 Elsevier B.V. All rights reserved.

KW - metric invariant

KW - systems

KW - automorphisms

U2 - 10.1016/j.physleta.2011.01.054

DO - 10.1016/j.physleta.2011.01.054

M3 - Article

VL - 375

SP - 1309

EP - 1318

JO - Physics Letters A

JF - Physics Letters A

SN - 0375-9601

IS - 10

ER -