Chaos in a nonautonomous eco-epidemiological model with delay

In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov's functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system's behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.

New existence results for nonlinear delayed differential systems at resonance

This paper deals with the first-order delayed differential systems { u ' + a ( t ) u = h ( t ) v + f ( t , u ( t - τ ( t ) ) ) , v ' + b ( t ) v = g ( t , u ( t - τ ( t ) ) ) , where a, b, τ, h are continuous ω-periodic functions with ∫ 0 ω a ( t ) d t = 0 and ∫ 0 ω b ( t ) d t > 0 ; f ∈ C ( R × [ 0 , ∞ ) , R ) and g ∈ C ( R × [ 0 , ∞ ) , [ 0 , ∞ ) ) are ω-periodic with respect to t. By means of the fixed point theorem in cones, several new existence theorems on positive periodic solutions are established. Our main results enrich and complement those available in the literature.

Existence of periodic solution for fourth-order generalized neutral p-Laplacian differential equation with attractive and repulsive singularities

In this paper, we investigate the existence of a positive periodic solution for the following fourth-order p-Laplacian generalized neutral differential equation with attractive and repulsive singularities: ( φ p ( u ( t ) - c ( t ) u ( t - δ ( t ) ) ) ″ ) ″ + f ( u ( t ) ) u ' ( t ) + g ( t , u ( t ) ) = k ( t ) , where g has a singularity at the origin. The novelty of the present article is that we show that attractive and repulsive singularities enable the achievement of a new existence criterion of a positive periodic solution through an application of coincidence degree theory. Recent results in the literature are generalized and significantly improved.

Complex dynamic behavior in a viral model with state feedback control strategies

With the consideration of mechanism of prevention and control for the spread of viral diseases, in this paper, we propose two novel virus dynamics models where state feedback control strategies are introduced. The first model incorporates the density of infected cells (or free virus) as control threshold value; we analytically show the existence and orbit stability of positive periodic solution. Theoretical results imply that the density of infected cells (or free virus) can be controlled within an adequate level. The other model determines the control strategies by monitoring the density of uninfected cells when it reaches a risk threshold value. We analytically prove the existence and orbit stability of semi-trivial periodic solution, which show that the viral disease dies out. Numerical simulations are carried out to illustrate the main results.

Dynamical behaviors for discontinuous and delayed neural networks in the framework of Filippov differential inclusions

This paper is concerned with the periodic dynamics of a class of delayed neural networks with discontinuous neural activation functions. Under the Filippov framework, the cone expansion and compression fixed point theorems of set-valued maps are successfully employed to derive the existence of the ω-periodic positive solution. However, before the discussion of the periodicity, there still remains a fundamental issue about viability to be solved due to the presence of general mixed time-delays involving both time-varying delays and distributed delays. This difficulty can be overcome by a transformation and the continuation theorem. Then, for the discontinuous and delayed neural network system with time-periodic coefficients, the uniqueness and global exponential stability of the periodic state solution are proved by using non-smooth analysis theory with generalized Lyapunov approach. Furthermore, the global convergence in measure of the periodic output is also investigated. The obtained results are a very good extension and improvement of previous works on discontinuous dynamical neuron systems with a broad range of neuron activations dropping the assumption of boundedness or monotonicity. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.