In this paper, we are interested in studying the oscillation of differential equations with a damping term and distributed delay. We establish new criteria that guarantee the oscillation of the third-order differential equation in terms of oscillation of the second-order linear differential equation without a damping term. By using the Riccati transformation technique and the principle of comparison, we obtain new results on the oscillation for the studied equation. The results show significant improvement and extend the previous works. Symmetry contributes to determining the correct methods for solving neutral differential equations. Some examples are provided to show the significance of our results.

In this paper, we introduce a new type of synchronization for the fractional order (FO) hyperchaotic models with different orders called compound-combination synchronization (CCS). Using the tracking control method, a theorem to calculate the analytical controllers which achieve our proposed synchronization is described and proved. We introduce, also, the FO hyperchaotic complex Lü, Chen, and Lorenz models with complex periodic forcing. The symmetry property is found in the FO hyperchaotic complex Lü, Chen, and Lorenz models. These hyperchaotic models are found in many areas of applied sciences, such as physics and secure communication. These FO hyperchaotic models are used as an example for our proposed synchronization. The numerical simulations show a good agreement with the analytical results. The complexity and existence of additional variables mean that it is safer and interesting to transmit and receive signals in communication theory. The proposed scheme of synchronization is considered a generalization of many types in the literature and other examples can be found in similar studies.

In this paper, we presented different behaviors such as chaotic and hyperchaotic of the generalized van der Pol oscillator with distributed order. We introduced the parameter intervals of these behaviors by computing the Lyapunov exponents of the oscillator, which is a good test for classifying the dynamical systems’ solutions. The active control approach with the Laplace transform technique was used to realize the antisynchronization and control of the proposed oscillator. Finally, numerical investigations have been carried out on the dynamics of the proposed oscillator to verify the reliability of our analytical results.

In this paper, we consider a certain class of third-order nonlinear delay differential equations with distributed arguments. By the principle of comparison, we obtain the conditions for the nonexistence of positive decreasing solutions as well as, and by using the Riccati transformation technique, we obtain the conditions for the nonexistence of increasing solutions. Therefore, we get new sufficient criteria that ensure that every solution of the studied equation oscillates. Asymmetry plays an important role in describing the properties of solutions of differential equations. An example is given to illustrate the importance of our results.

This paper introduces the chaotic fractional-order Sprott Q system and its dynamics. The double compound combination of eight fractional-order chaotic synchronizations was investigated. This kind of synchronization is considered a generalization of many types of synchronization given in the literature. The analytical formula of the control functions is developed and proven using the tracking control method. As we know the fractional derivatives of two multiple functions are very difficult to calculate, and the tracking control method is more suitable for this kind of synchronization. This technique’s communication is more secure and reliable because there are more drive and response systems. To achieve the proposed synchronization, four driving and four response identical chaotic systems are used as an example. This scheme may be used in many applications such as secure communications and safe information. Using the proposed double compound-combination synchronization is an example given to encrypt a message. The analytical results were confirmed by numerical simulation, and we found good agreement.