^{1,a)}and Elbert. E. N. Macau

^{2,b)}

### Abstract

In this work, we propose an adaptive node-to-node pinning control strategy. In this approach, both the coupling strength among nodes and the pinning control gains are adaptively changed according to well chosen adaptation laws that take into account the specificities of the oscillators and the network topology. Proof of stability and performance comparison is also shown in this paper.

A great variety of nowadays systems can be regarded as networks of interconnected chaotic dynamical agents, i.e., oscillators which interact with each other. Using this network description, the oscillators are located in nodes, while the edges of the network capture the interaction links among the oscillators. Also, researchers in different areas of applied science and engineering have addressed the problem of how to apply control strategies that exploit the network topology and the information exchange among the nodes to obtain desired ordered collective behaviors in the interconnected system. Examples of these control actions necessarily include rendezvous and flocking problems in robotics,

^{1}synchronization of sensornetworks,

^{2–5}consensus and multi-agent coordination problems in control theory,

^{6,7}the emergence of coordinated motion in animal behavior, and other biological systems (see, for instance, Refs. 8 and 9). In this work, we propose an adaptive node-to-node pinning control strategy to achieve control goals in distributed scenarios.

We would like to thank FAPESP and CNPq for the financial support to this work.

I. INTRODUCTION

II. ADAPTIVE PINNING CONTROL STRATEGY

A. Proof of Theorem 1

III. NUMERICAL RESULTS

IV. CONCLUSION

### Key Topics

- Synchronization
- 11.0
- Oscillators
- 10.0
- Networks
- 7.0
- Network topology
- 5.0
- Numerical solutions
- 5.0

## Figures

Pinning error : Figure shows the evolution of the pinning synchronization error of each state variable of each node of the network. Observe states' error go to zero as time evolves. In the figure, each different color represents one node different node of the network.

Pinning error : Figure shows the evolution of the pinning synchronization error of each state variable of each node of the network. Observe states' error go to zero as time evolves. In the figure, each different color represents one node different node of the network.

Synchronous state: Figure shows the time evolution of each state variable of each node of the network. Observe states go to the equilibrium point set by the reference node as time evolves. In the figure, each different color represents one node different node of the network.

Synchronous state: Figure shows the time evolution of each state variable of each node of the network. Observe states go to the equilibrium point set by the reference node as time evolves. In the figure, each different color represents one node different node of the network.

Coupling strength : Figure shows the evolution of each coupling strength of the network edges (links). Observe each coupling strength converges to a bounded value as time evolves. In the figure, each different color represents one node different coupling strength .

Coupling strength : Figure shows the evolution of each coupling strength of the network edges (links). Observe each coupling strength converges to a bounded value as time evolves. In the figure, each different color represents one node different coupling strength .

Pinning control gain *q*(*t*): Figure shows the evolution of the pinning control gain of the pinned node *j*. Observe each control gain converges to a bounded value as time evolves. In the figure, each different color represents the control gain of a different pinnable node.

Pinning control gain *q*(*t*): Figure shows the evolution of the pinning control gain of the pinned node *j*. Observe each control gain converges to a bounded value as time evolves. In the figure, each different color represents the control gain of a different pinnable node.

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