## Abstract

Swarmalators, which combine the swarming phenomenon with synchronization, have drawn a lot of attention from researchers in recent years. Nevertheless, the majority of earlier research contends with identical swarmalators, thus neglecting dynamics that might emerge with parameter heterogeneity. In this paper, we therefore study the internal dynamics of non-identical swarmalators with natural frequencies sampled from six distributions: uniform, Gaussian, Lorentz, and its three generalizations. We show how these different distributions relate to the spatial dynamics and how they affect the phase transitions between different dynamical states. We find that, in comparison to the other five distributions, the phase transition curve for the uniform frequency distribution exhibits a more explosive character. Moreover, by changing the phase coupling strength, we find three distinct spatial patterns: drifting periodic, irregular, and static. With these results, we improve our understanding of the complex interplay between synchronization and swarming in non-identical swarmalators, and we also hope to inspire further research along the same lines.

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## Data Availability

Data generated during the current study will be made available on reasonable request.

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## Acknowledgements

We would like to thank Dr. Fatemeh Parastesh for all her comments during this research.

## Funding

M.P. was supported by the Slovenian Research and Innovation Agency (Javna agencija za znanstvenoraziskovalno in inovacijsko dejavnost Republike Slovenije) (Grant Nos. P1-0403).

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### Contributions

Conceptualization: Fahimeh Nazarimehr, Farnaz Ghassemi, Dibakar Ghosh; Methodology: Matjaž Perc, Dibakar Ghosh; Formal analysis and investigation: Sheida Ansarinasab, Fahimeh Nazarimehr, Gourab Kumar Sar; Writing-original draft preparation: Sheida Ansarinasab, Fahimeh Nazarimehr, Gourab Kumar Sar; Writing-review and editing: Farnaz Ghassemi, Dibakar Ghosh, Sajad Jafari, Matjaž Perc; Resources: Farnaz Ghassemi, Dibakar Ghosh; Supervision: Farnaz Ghassemi, Matjaž Perc, Dibakar Ghosh.

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## Natural frequency distributions

### Natural frequency distributions

Here, we briefly discuss the mathematical forms of the chosen distributions for the swarmalators’ natural frequencies.

### 1.1 Uniform distribution

The continuous uniform distribution is a probability distribution widely employed in statistical analysis and modeling to describe phenomena exhibiting equal likelihood within a defined interval [52]. Denoted as *U*(*e*, *f*), with *e* and *f* representing the lower and upper bound, respectively, this distribution maintains a constant probability density function (PDF) across the interval. The PDF for the continuous uniform distribution is as follows,

where \(g(\omega )\) shows the probability density at a given frequency \(\omega \). This distribution is characterized by a uniform probability of occurrence for all values within the interval, resulting in a flat PDF curve. In this work, we consider the uniform distribution of the swarmalators’ natural frequencies in the interval of \((-0.1,0.1)\).

### 1.2 Standard Gaussian distribution

The standard Gaussian distribution represents a fundamental probability distribution utilized in various scientific disciplines. It is a continuous distribution characterized by a symmetrical bell-shaped curve [53]. The standard Gaussian distribution is parameterized by a zero mean and unit standard deviation. The PDF for this distribution is given by,

In Eq. (7), \(g(\omega )\) displays the density at a given value \(\omega \). The parameters \(\omega '\) and \(\sigma \) represent the mean and standard deviation of natural frequencies, respectively. These parameters are selected as \(\omega '=0\) and \(\sigma =1\) in standard Gaussian distribution. This distribution exhibits intriguing properties, providing a crucial framework for understanding and analyzing various phenomena in real-world applications [54, 55].

### 1.3 Generalized Lorentz distribution

The generalized Lorentz distribution, also called the generalized Cauchy distribution, is a probability distribution that is particularly useful in cases where phenomena exhibit long-tailed behavior, capturing the occurrence of extreme events [56]. Extending the classic Lorentz distribution, the generalized version introduces an additional degree parameter *n* to enhance its flexibility in tail modeling [17]. The distribution is characterized by its PDF, given by the following formula,

where \(g(\omega )\) represents the density at a given natural frequency \(\omega \). The parameter \(\omega _0\) denotes the central natural frequency where the peak of the PDF occurs, and \(\varDelta \) controls the scale. The generalized Lorentz distribution stands out due to its ability to capture long tails, indicating a higher likelihood of extreme values than the Gaussian distribution. Without loss of generality of the problem, we use the standard form of the distribution by selecting \(\omega _0=0\) and \(\varDelta =1\). In this study, the generalized Lorentz distribution is applied for n=1,2,3, and 4. By substituting \(n=1\) into Eq. (8), the conventional Cauchy-Lorentz distribution is obtained. The probability distribution functions diagrams of the uniform, standard Gaussian, Lorentz distribution and its generalizations are presented in Fig. 14. These diagrams show that as the degree *n* of the generalized Lorentz distributions increase, the corresponding functions become flatter with thinner tails.

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Ansarinasab, S., Nazarimehr, F., Sar, G.K. *et al.* The spatial dynamics and phase transitions in non-identical swarmalators.
*Nonlinear Dyn* **112**, 10465–10483 (2024). https://doi.org/10.1007/s11071-024-09625-5

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DOI: https://doi.org/10.1007/s11071-024-09625-5