The paper deals with the design of backgrounds to construct novel chaotic systems grounded on the known ones. These backgrounds are based on using variable-sign functions to replace linear dependencies between dynamical systems’ state variables and replacing time-depended functions with differential equations, which solutions define the above-mentioned time-depended functions. Such an approach allows us to consider a wide range of dynamical systems whose motions are caused by external harmonic excitation signals produced by linear oscillators. We modify the linear exciter using nonlinear feedbacks to produce non-symmetrical nonlinear oscillations. Such feedbacks are defined using variable-sign and constant-sign power functions of exciter state variables with integer and non-integer power factors. Contrary to the classical approach, which uses only linear algebraic feedforward between exciter and dynamical system, we offer to use various integro-algebro-differential links to make the system dynamic more complex by transforming exciter output in time domain. Also, we offer to apply these links to drive a dynamical system by exciter and vice versa. Our studies show that, in this case, both the dynamical system and the exciter can have chaotic dynamics. We offer to use such an approach to consider the chaotic system as two interconnected chaotic subsystems and improve the secured features and performances.