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Nonlinear dynamic performances such as homoclinic bifurcation and chaos were investigated for tristable vibration energy harvesting systems. The analytical expression of the symmetric and asymmetric homoclinic solution was obtained through the Padé approximation, which was consistent with the numerical solution. According to the Melnikov theory, the qualitative method of studying the homoclinic bifurcation of the energy harvesting system with a triple well was developed, and the necessary condition for the occurrence of homoclinic bifurcation was obtained. Numerical simulations yielded bifurcation diagrams and maximum Lyapunov exponents that demonstrated the inter-well responses predicted with the Melnikov method. Compared with the system with symmetric potential energy, the system with asymmetric potential energy has a lower threshold of homoclinic bifurcation. For a low excitation level, the system with asymmetric potential energy witnesses inter-well chaos, while the response of the system with symmetric potential energy still keeps trapped in a single well. The change of symmetry of the system potential energy function improves the output voltage due to the increase in the probability of generating a large periodical inter-well oscillation response. The research on the homoclinic bifurcation of nonlinear energy harvesting systems with symmetric and asymmetric triple potential wells provides an effective tool for the parametric design of high-performance energy harvesters. [ABSTRACT FROM AUTHOR]

The asymmetry and non-smooth factors of the unilaterally constrained shape memory alloy beam model are discussed to influence the chaotic motion of the system. By studying the smooth Melnikov function and the non-smooth Melnikov function coexisting with the nonlinear vibro-impact system, the necessary conditions for Smale horseshoe chaos generated by the smooth homoclinic and non-smooth homoclinic bifurcations are obtained. The numerical simulation results such as phase diagram, Poincare cross-sectional diagram, bifurcation diagram and maximum Lyapunov exponent are used to verify the chaotic threshold conditions. The results show that: The larger harmonic force is conducive to promoting bifurcation of the system's homoclinic orbit to generate chaos; the interaction of smooth homoclinic bifurcation and non-smooth homoclinic bifurcation can induce the occurrence of chaotic mergers. [ABSTRACT FROM AUTHOR]

Investigating codimension two bifurcations in neutral differential-difference equations may reveal the complicated dynamics. In this paper, center manifold reduction method is established to derive the normal forms near the codimension two bifurcations in NDDE. We first introduce the decomposition of the phase space of NDDE. Then the normal form method to reduce a NDDE onto an invariant center manifold is obtained. Finally BT bifurcation in van de Pol's equation with extended delay feedback is considered and the detailed dynamics, including the globally existed homoclinic orbits, are obtained near the equilibrium. [ABSTRACT FROM AUTHOR]