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The network anomaly detection is the concept of analyzing network traffic in the search of any sign of "abnormal behavior" because such anomalies are often symptoms of network faults or security breaches. The main aim of this paper is to investigate the dynamics of address resolution protocol (ARP) traffic time series using Fourier and nonlinear methods. The experimental network consists of workstations connected through several network switches and one router with Internet connection. Two cases of workstation traffic are analyzed: minimal and continuous. It is shown that the ARP traffic has long time period of 23.5 hours and short time periods in two intervals (2-6) and (9-12) minutes. These periods are proved to be independent of the workstation traffic intensity. The time series of the ARP frames whereas has deterministic chaos character. The largest Lyapunov exponent indicates that the time of stability loss of ARP time series is equal to 15 minutes.