Multimode Oscillations in Coupled Oscillators With High-Order Nonlinear Characteristics

Abstract

We propose two van der Pol oscillators with high-order nonlinearities coupled with an inductor and investigate various types of multimode oscillations. We confirm that these oscillations are stably excited by numerical calculations, theoretical analyses and circuit experiments. In particular, we aim to show relations between the multimode oscillations and the number of order nonlinear characteristics.

Analysis and Design of Dual-Mode CMOS LC-VCOs

“…Thus, the linear system is stable if (21) which hold under the same assumption of negligible losses, used for deriving and in (6). As and , from (20) and (21) we obtain the stability condition for a limit cycle (22) which can be verified by and by for some values of the circuit parameters, but cannot be satisfied by the solution , because and, hence, and . The superposition of the two diagrams of the amplitudes of oscillations and , shown in Fig.…”

“…Starting from low values of , according to (22) a stable oscillation exists in the circuit even when, increasing , the circuit enters the region where both the oscillations at frequency and are possible. The amplitude (frequency) of the oscillation steadily decreases up to where (22) ceases to be fulfilled and the circuit abruptly passes into the operation mode where oscillations with frequency and amplitude are generated. In the reverse direction, i.e., decreasing , the circuit enters the overlap region from the side where an oscillation with frequency and amplitude takes place.…”

“…That oscillator, with some variant of the additional circuit, was later studied by using the method of regular perturbation [12], [13], the describing function method [14], [15], and the averaging method of Krylov and Bogoliubov [16]- [18]. This latter method was applied in [19]- [21], for seeking the conditions for the simultaneous occurrence of two oscillation modes and, more recently, in [5], [22].…”

Dynamical Behaviors of Coupled Memristor-Based Oscillators with Identical and Different Nonlinearities

“…This problem appears in many applications in biology [2], neuroscience [3][4][5][6], chemistry [7,8], and physics [9][10][11]. Researchers in the field of dynamical systems are interested in studying the stability of coupled oscillators systems with nonlinear properties since 1950s till now [12][13][14][15].…”