Synched is a piece of software created to visualize synchronization phenomena in complex systems. Due to their ubiquity in science and nature, the study of large networks of coupled oscillators has become a popular research area. Phase oscillator systems such as the Kuramoto model have become paradigmatic examples for studying synchrony in such systems. Synched provides an easy and intuitive way to visualize the dynamics of these systems in real-time. Below you will find a link to download Synched as well as instructions on how to use it. For further information on my research, including publications, check out my home page.

Synched is free to download for non-commercial purposes, under the Creative Commons license linked above. The current version is 1.0 and has been tested on Mac OSX 10.6 and 10.7. I plan to release a Linux version in the future. I would like to thank my colleague Dan Larremore for his guidance, as well as my advisor Juan G. Restrepo and group member Dane Taylor for their ideas and comments. Finally, funding from NSF Grant No. DMS-0908221 and the University of Colorado at Boulder's Department of Applied Mathematics.

## How to use Synched

Here is a sample screenshot of Synched in action. The window is divided into 2 parts: the display (left panel) and the controls (right panel). Each panel is discussed in detail below.

### Left Panel - The Display

The display is organized into two areas: an outer circle and an inner circle.

The outer circle depicts the information for each individual oscillator. Along the outer circle the phase of each individual oscillator is plotted as a small circle. For the sake of discerning oscillators with very similar phase, the radius with which each oscillator is plotted is slightly shuffled. Also, by clicking at any point on the screen you can force oscillators to a particular phase with a forcing strength proportional to F (described below).

The inner circle depicts the collective macroscopic information of the system. Each model included in Synched has one or more order parameters that describe the degree of synchrony in the system. These order parameters are complex numbers in the unit circle. The larger the magnitude, the more synchronized the system is. For more information of the meaning of order parameters for the different systems, see the model description.

Furthermore, information about the model as well as degrees of synchrony are shown in the corners. This information can be enabled or disabled in the control panel.

### Right Panel - The Controls

Master Controls:

• Model allows you to change the model for your next simulation. You can choose between the Kuramoto, Time-Delayed, Clusters, and Communities. For specifics of each model, see the model description.

• # Oscillators allows you to change the number of oscillators for your next simulation. You can choose between 49, 100, 400, and 900.

• New Simulation resets the system to simulate the model chosen above with the number of oscillators chosen above.

Parameters Controls: Depending on the current model being simulated, only some of the parameters are used. Unused parameters are gray-ed out and disabled.

• k changes the local coupling strength. This parameter is only used for the Communities model. The local coupling strength is the coupling strength between oscillator within the same community

• K changes the global coupling strength in the system. This parameter is used in all models.

• Mean frequency changes the mean of all intrinsic frequencies. This parameter is used in all models.

• Cluster q changes the clustering number of the coupling function. This parameter is only used for the Clusters model. The clustering number describes which mode couples oscillator together.

• Forcing F changes strength of forcing on the system. By clicking and holding, you can force oscillators to a particular phase with a forcing strength proportional to F.

Display Controls:

• Display Info. toggles the information visible/invisible. Model and synchrony information are visible when checked, invisible when unchecked.

• Show Colors toggles the color option on/off. Oscillators are colored accordingly when on and white when off.

• Speed sets the speed of the simulation. In order to observe nuances of the transient dynamics, for instance, choose a slower speed. If you would like to "fast-forward" to a steady-state, you can quickly bypass the transient state by choosing a fast speed.

Quit closes the program.