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A nonlinear equation with piecewise continuous argument. (English) Zbl 0723.34061

Summary: Asymptotic and qualitative behavior of solutions is established for the equations \((1)\quad x'(t)=\mu x(t)(1-x([t])),\) \((2)\quad x'(t)=\mu x(t)(1-x(2[(t+1)/2])),\) where \(\mu\) is a positive parameter. Comparison is made with the continuous logistic equation \((3)\quad x'(t)=\mu x(t)(1- x(t))\) and the discrete logistic equation \((4)\quad x_ n=\mu x_{n- 1}(1-x_{n-1}).\) One result is that (1) and (4) can exhibit complicated dynamics and (2) and (3) cannot.

MSC:

34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
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