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The precise nature of TCP dynamics over Internet connections has not been well understood, since the existing results are solely analytical or based on simulations. We employ the time-dependent exponent curves and logarithmic displacement curves to study TCP AIMD congestion window-size traces over Internet connections. We show that these dynamics have two dominant parts, a stochastic component in response to network traffic and a deterministic chaotic component due to the nonlinearity of protocol. These dynamics can be largely characterized as anomalous diffusions with a very large exponent.