We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations. These equations generalize Liénard equations and are used to describe self-sustained oscillations. Our results are valid for arbitrary degrees of the polynomials arising in the equations. We find a number of novel Liouvillian integrable subfamilies. We derive an upper bound with respect to one of the variables on the degrees of irreducible Darboux polynomials in the case of nondegenerate or algebraically degenerate near infinity polynomial Levinson–Smith equations. We perform the complete classification of Liouvillian first integrals for the nondegenerate or algebraically degenerate near infinity Rayleigh–Duffing–van der Pol equation that is a cubic Levinson–Smith equation.

This paper designs a locally active memristor with two variable parameters based on Chua’s unfolding theorem. The dynamical behavior of the memristor is analyzed by employing pinched hysteresis loop, power-off plot (POP), DC V–I curve, small-signal analysis, and edge-of-chaos theory. It is found that the proposed memristor exhibits nonvolatile and bistable behaviors because of coexisting pinched hysteresis loops. And the variable parameters can realize the rotation of the coexisting pinched hysteresis loops, regulate the range of the locally active region and even transform the shape of the DC V–I curve into S-type or N-type. Furthermore, a simple oscillation circuit is constructed by connecting this locally active memristor with an inductor, a capacitor, a resistance, and a bias voltage. It is shown by analysis that the memristive circuit can generate complex nonlinear dynamics such as multiscroll attractor, initial condition-based dynamics switching, transient phenomenon with the same dynamical state but different offsets and amplitudes, and symmetric coexisting attractors. The measurement observed from the implementation circuit further verifies the numerical results of the oscillation circuit.

Vector-borne disease malaria is transmitted to humans by arthropod vectors (mosquitoes) and contributes significantly to the global disease burden. TV and social media play a key role to disseminate awareness among people by broadcasting awareness programs. In this paper, a nonlinear model is formulated and analyzed in which cumulative number of advertisements through TV and social media is taken as dynamical variable that propagates awareness among people to control the prevalence of vector-borne disease. The human population is partitioned into susceptible, infected and aware classes, while the vector population is divided into susceptible and infected classes. Humans become infected and new cases arise when bitten by infected vectors (mosquitoes) and susceptible vectors get infected as they bite infected humans. The feasibility of equilibria is justified and their stability conditions are discussed. A crucial parameter, basic reproduction number, which measures the disease transmission potentiality is obtained. Bifurcation analysis is performed by varying the sensitive parameters, and it is found that the proposed system shows different kinds of bifurcations, such as transcritical bifurcation, saddle-node bifurcation and Hopf bifurcation, etc. The analysis of the model shows that reduction in vector population due to intervention of people of aware class would not efficiently reduce the infective cases, rather we have to minimize the transmission rates anyhow, to control the disease outbreak.

In this paper, we consider a Holling type II predator–prey system with prey refuge, Allee effect, fear effect and time delay. The existence and stability of the equilibria of the system are investigated. Under the variation of the delay as a parameter, the system experiences a Hopf bifurcation at the positive equilibrium when the delay crosses some critical values. We also analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution by the center manifold theorem and normal form theory. We show that the influence of fear effect and Allee effect is negative, while the impact of the prey refuge is positive. In particular, the birth rate plays an important role in the stability of the equilibria. Examples with associated numerical simulations are provided to prove our main results.

For a class of nonlinear diffusion–convection–reaction equations, the corresponding traveling wave systems are well-known nonlinear oscillation type of systems. Under some parameter conditions, the first integrals of these nonlinear oscillators can be obtained. In this paper, the bifurcations, exact solutions and dynamical behavior of these nonlinear oscillators are studied by using methods of dynamical systems. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink and anti-kink wave solutions, as well as limit cycles, are obtained. Most important and interestingly, a new global bifurcation phenomenon of limit bifurcation is found: as a key parameter is varied, so that singular points (except the origin) disappear, a planar dynamical system can create a stable limit cycle.

Vibration impact is often used in the piezoelectric energy harvesting (PEH) system to increase the effective bandwidth of the harvester. Viscoelastic materials have been used successfully to mitigate vibration problems in various types of mechanical systems such as buildings, cars, aircraft and industrial equipment. However, less research has been done on the energy harvesting system with impact and viscoelastic force driven by random excitation. Stochastic response of an impact PEH system with viscoelastic force under Gaussian white noise excitation is investigated in this paper. Firstly, by transforming the variables, viscoelastic force can be substituted with the stiffness and damping terms to get an approximately equivalent system without viscoelastic term. Secondly, the approximate analytical solutions are acquired by the stochastic averaging method and nonsmooth coordinate transformation. The validity of this theoretical approach is confirmed by comparing the analytical solutions with the numerical solutions derived from the Monte Carlo method. Then, the effect of noise intensity and nonlinear damping coefficient on the stochastic response of the system is discussed. It is concluded that the restitution coefficient, viscoelastic component, relaxation time and linear damping coefficient can induce the occurrence of stochastic P-bifurcation. Finally, the roles of system parameters on the mean square voltage and average output power of the energy harvester are investigated respectively.

The Investment Savings-Liquidity preference Money supply (IS-LM) model is represented as a graph depicting the intersection of products and the money market. It elaborates how an equilibrium of money supply versus interest rates may keep the economy in control. In this paper, we combine the basic business cycle IS-LM model with Kaldor’s growth model in order to create an augmented model. The IS-LM model, when coupled with a certain economics expansion (in our instance, the Kaldor–Kalecki Business Cycle Model), provides a comprehensive description of a developing but robust economy. Right after the introduction of capital stock into the system, it cannot be employed and also, while making some investment choices, this requires some time in execution, which ultimately alters resources, i.e. capital. Thus, in the capital accumulation, we will be incorporating double time delays in Gross product and Capital Stock. These time delays represent the time periods during which investment decisions were made and executed and the time spent in order for the capital to be put to productive use. After formulating a mathematical model using delayed differential equations, dynamic functioning of the system around equilibrium point is examined where three instances appeared based on time delays. These cases are: when both delays are not in action, when only one delay is in action and when both delays are in action. It is shown that time delay affects the stability of the equilibrium point and, as the delay crosses a critical point, Hopf bifurcation exists. It is observed that by using Kaldor type investment function, the delay residing in capital stock only will destabilize in less time as compared to when both the delays are present in the system. The system is sensitive to certain parameters which is also analyzed in this work.