Following a research line opened in a previous work, this paper aims to describe in the language of kinetic theory the evolution of systems consisting of a very large number of particles interacting not only with each other but also with the external world, so that the equation governing their evolution contains an additional term representing such interaction, called the “forcing term”. Notice that such a term does not express a “classical” force exerted by the external world on the particles, but a more general influence on the effects of mutual interactions of particles. In our everyday experience, indeed, several phenomena external to well-defined systems, if properly interpreted, can be described as “forcing terms” that modify the rules of interactions between the elements of the system: for instance, climate changes that increase or decrease the different agricultural productions at different times, thus altering the economic relationships between different subsystems, which in turn can also be perturbed by stock market fluctuations, sudden wars, periodic epidemics, and so on. Thus, in connection with these problems, the mathematical analysis of the effects of different kinds of forcing terms on the solutions to the equations governing the collective (that is, statistical) behavior of such a many-particle system seems to be of some interest. In the present paper we offer a study of the basic mathematical properties of such solutions in the special but particularly interesting case in which the forcing term is assumed to depend on the history of the distribution of states. This dependence is taken to depict the ability of individuals to learn from past experiences, so that the reactions to external changes are different for different past experiences. Some numerical simulations to show the effects of forcing terms in some particular cases conclude the paper.
One of the main problems studying the dynamics of a polynomial differential system in the plane is to determine the non-existence, existence and its number of limit cycles. Here we study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of cubic polynomial centers ẋ=x ( x^2-y^2+2 x+α ^2+1) , ẏ=y ( x^2-y^2-2 y-α ^2-1) , where α 0 , when it is perturbed inside the classes of all cubic polynomial differential systems. We obtain by using the averaging method of first order that the maximum number of limit cycles is 3 and it is reached. The fact that the unperturbed center has the two invariant straight lines x=0 and y=0 can be the reason that the number of limit cycles of the perturbed systems be so small respect other cubic polynomial differential systems.
We investigate a stochastic delayed behavioral epidemic model in which population awareness is governed by past infection prevalence. The model is formulated as a stochastic delay differential equation with multiplicative Itô noise. By analyzing the logarithmic growth rate of the infective class, we establish a sufficient extinction condition in terms of the long-term averaged logarithmic drift. We consider a heuristic stochastic threshold that summarizes the effect of environmental noise near the disease-free regime. We also clarify the additional pathwise estimates that would be required to obtain a rigorous stochastic persistence threshold; mean boundedness alone is not sufficient. The results show that environmental noise can suppress infection transmission and drive the disease to extinction. Numerical simulations corroborate the theoretical findings and illustrate that extinction may occur even when the corresponding deterministic model predicts persistent infection.
This research introduces a heterogeneous memristive neuron network coupling a discrete Rulkov neuron and a discrete Aihara neuron through a second-order leaky memristor. Its two internal states, driven separately by the neurons, emulate a synaptic channel and an electromagnetic-radiation-like field. The resulting high-dimensional MRA map exhibits bounded hyperchaos with positive Lyapunov exponents and hidden attractors. An ablation study confirms that each of the three memductance nonlinearities (sinusoidal, hyperbolic-tangent, bilinear) contributes distinctly to hyperchaos, and their superposition is necessary. Dynamical analysis reveals hyperchaos, multistability, and diverse firing patterns. Cryptographic suitability is confirmed by the full NIST test suite, a chi-square uniformity test, and key-sensitivity, differential, noise, and shear tests. An image encryption algorithm derives its chaotic keystream from the MRA map, while the secret is a user-supplied external key combined with the plaintext hash. This resolves the circularity of purely plaintext-derived keys and provides a rigorous, non-inflated key space independent of the plaintext. Bit-level permutation via chaos-based row-column cycle shift scrambling is followed by chaotic sequence-based diffusion. The scheme achieves uniform histograms, negligible correlation, high entropy, and resistance to differential attacks. Efficiency is assessed via theoretical complexity, with runtime figures qualified as literature-reported and hardware-dependent. This interdisciplinary work bridges bio-inspired neuronal dynamics and chaos-based cryptography for secure applications and neuromorphic computing.
This article examines the dynamical behavior of a fractional-order allelopathic plankton model subject to harvesting. The governing equations are expressed in terms of Caputo fractional derivatives with time-dependent delays. Utilizing the Banach fixed-point theorem, we derive explicit sufficient conditions ensuring the existence and uniqueness of a weighted pseudo S-asymptotically ω -periodic solution confined to an invariant set. Moreover, local Mittag-Leffler stability is established via a Lyapunov functional method combined with fractional order inequalities. The theoretical findings are supported by numerical simulations that demonstrate the applicability of the proposed framework.
