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Both capacitor and charge-controlled memristor (CCM) can regulate the inner energy levels during the release and accumulation of charges in these electronic components. Combination of inductive and capacitive components is crucial to build an oscillatory circuit, and incorporation of a nonlinear resistor is helpful to support nonlinear oscillation under continuous energy exchange between magnetic field and electric field. Relation between physical variables in neural circuits can be described by some equivalent nonlinear oscillators by applying scale transformation on the variables and parameters for the neural circuit, and the energy function becomes available from theoretical approach. In this work, a simple neural circuit coupled with a CCM is designed to obtain a memristive neuron model, and then linear transformation is applied to the sampled variables involved with time scale for time step, a new memristive map is obtained to discuss its energy characteristic and self-controllability under energy flow. The result provides guidance to measure the reliability of maps, and the new derived maps with reliable physical evidences can be further used in digital signal processing. • Memristive neuron and memristive map are presented. • Linear transformation from oscillator to equivalent map is presented. • Energy function for a memristive map is defined. • Adaptive law is proposed to activate mode transition and energy shift in the map neuron. [ABSTRACT FROM AUTHOR]