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In this paper, a new generalized nonlinear Schrödinger (GNLS) equation is investigated by Darboux matrix method. Firstly, the n-fold Darboux transformation (DT) of the GNLS equation is constructed. Then, the soliton solutions, breather solutions, and rogue wave solutions of the GNLS equation are studied based on the DT by choosing different seed solutions. Furthermore, the dynamic features of these solutions are explicitly delineated through some figures with the help of Maple software.

This study exposes the analytical and numerical analyses of multistable systems for energy harvesting purposes. More specifically, this paper aims at providing appropriate conditions for ensuring equal potential barrier to go from one well to another whatever the order of multistability. This therefore allows optimal operations through either potential barrier lowering or vibration magnitude increase. Then, such analytical and numerical results are incorporated into a general dynamic model and evaluated. Results show a significant magnification of the frequency bandwidth while keeping the same maximal velocity magnitude. Hence, such a unified approach would permit designing efficient vibrational energy harvester working on a wide frequency band at low excitation levels.

Numerical continuation tools are nowadays standard methods for the bifurcation analysis of dynamical systems. Unfortunately, the full power of these methods is still unavailable in experiments, in particular, if no underlying mathematical model is at hand. We here aim to narrow this gap by providing control based continuation of periodic states which can be ultimately implemented in real-world experimental set-ups. Taking inspiration from atomic force microscopy, we develop experimentally relevant control and tracking tools for time periodic solutions in driven nonlinear oscillator systems based on stroboscopic maps.

In view of the fact that dissipative chaotic systems contain attractors in phase space and are easy to reconstruct, resulting in security flaws in encryption. We propose to design a conservative chaotic system with controlled chaotic scrolls, and in phase space, there is no attractor, and the direction and the number of chaotic scrolls can be controlled. The ergodic space in phase space becomes richer as the number of scrolls increases, com,plicating the chaotic behavior. Based on this conservative chaotic system, we present a novel color image encryption algorithm. To make the mixing of each plane of the color image more sufficient, we invented a plane element rearrangement and a dynamic selection row–column cross scrambling method in this algorithm. A cross-plane diffusion method was also created to make each plane part of the image form a whole, and each plane element will change as a result of the modification of one element. According to experimental and analysis results, the color image encryption technique offers strong security and real-time communication.

This work establishes the heteroclinic tangles when the Melnikov function is degenerate. By deriving the higher order Melnikov functions, we obtain an explicit formula of the second-order Melnikov function $$E_1(t_0)$$ , which essentially facilitates the evaluation and application to the concrete differential equations. Two examples manifest this efficient method to characterize chaotic heteroclinic tangle.

To study the control problem of vehicle following system with a rational initial state is of greater practical significance than that with an irrational initial state. The Petri net is used to describe the state transition diagram of vehicle following control system with a rational initial state after vehicle following control model is constructed. An integrated control method, including the velocity-difference control based on dynamic safety following distance, the inter-vehicle distance control for efficiency improvement based on hyperbolic function and the emergency stopping control for collision avoidance, under their respective conditions, is presented for the establishment of a safe and efficient steady-following state. The numerical results show that the presented control method can be used by the following vehicle to adapt its own behavior in safety and efficiency to the preceding vehicle’s behavioral change, and lead vehicle following system with a rational initial state to enter the previous safe and efficient steady-following state or a new safe and efficient steady-following state.