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I am reading the book "Multiple input describng functions and nonlinear system design" written by A.Gelb and W.E.Vander Velde.

At some point it says:

Single-valued characteristics are termed memoryless; multivalued characteristics are said to possess memory.

I am a bit confused about the meaning of the 'memoryless' term. I thought that memoryless meant a function that depends only on the value at that time instant and not at previous instants of time.

What am I missing?

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The definition as you quoted it is only correct for static nonlinearities. The output of a static nonlinearity only depends on the input function directly, and not on its integral or derivatives. Otherwise, the nonlinearity is called dynamic (i.e. having memory). A static nonlinearity can also have memory if it is multi-valued, i.e. if there is more than one possible output value for a given input value (e.g. a system with hysteresis). Finally, a static nonlinearity is indeed memoryless if it is single-valued, i.e. if its output is uniquely defined by its input value:

$$y(t) = F(x(t))$$

where $y=F(x)$ is an injective function (i.e. each $x$ is mapped to only one value $y$).

So in sum you have

  • dynamic nonlinearity: has memory
  • static nonlinearity
    1. multi-valued: has memory
    2. single-valued: memoryless
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