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In this work, the dynamic of the triode-based Van der Pol oscillator coupled to a linear circuit is investigated (Triode-based VDPCL oscillator). Towards this end, we present a mathematical model of the triode, chosen from among the many different ones present in literature. The dynamical behavior of the system is investigated using classical tools such as two-parameter Lyapunov exponent, one-parameter bifurcation diagram associated with the graph of largest Lyapunov exponent, phase portraits, and time series. Numerical simulations reveal rather rich and complex phenomena including chaos, transient chaos, the coexistence of solutions, crisis, period-doubling followed by reverse period-doubling sequences (bubbles), and bursting oscillation. The coexistence of attractors is illustrated by the phase portraits and the cross-section of the basin of attraction. Such triode-based nonlinear oscillators can find their applications in many areas where ultra-high frequencies and high powers are demanded (radio, radar detection, satellites communication, etc.) since triode can work with these performances and are often used in the aforementioned areas. In contrast to some recent work on triode-based oscillators, LTSPICE simulations, based on real physical consideration of the triode, are carried out in order to validate the theoretical results obtained in this paper as well as the mathematical model adopted for the triode.

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Nowadays, we are witnessing a dramatic advance in wireless technology-based magnetic induction. It is used both for wireless power transfer and data transfer between systems. In addition, it is widely shown that a network of coupled identical oscillators exhibits complex collective behavior characterized by the coexistence of coherent and incoherent domains and termed as chimera state. In this paper, we consider a network of (N ≥10) locally and magnetically coupled Van der Pol oscillators coupled to a linear circuit (VDPCL oscillators). We then investigate the different arrangements of their interactions in terms of the magnetic coupling coefficients, taken as the bifurcation parameters. Statistical measure namely the strength of incoherence is used to classify the synchronized states in the network. Another algorithm described in the text is used for the classification and is consistent with the strength of incoherence. Numerical simulation reveals that the emerging spatiotemporal behaviors depend on the choice of initial conditions revealing the presence of multistability in the network. This network configuration also reveals a rich repertoire of spatiotemporal dynamics such as coherence/global synchronization, decoherence, chimera state, cluster synchronization, and solitary states as the magnetic coupling coefficients vary. Some other interesting behaviors such as traveling clustered wave, double and multicluster chimera state, and clustered solitary state for a specific set of initial conditions are also obtained. Furthermore, Pspice-based simulations carried out for a network of (N=10) oscillators are consistent with the numerical simulations based on the mathematical model.