Study of internal oscillations in a dynamic system through an external signal implemented in circadian rhythms

Through the analysis carried out on a dynamic model that is represented as a system of ordinary differential equations that describes the behavior of the circadian cycles; we will show and analyze in the next document what are the conditions that allow the synchronization of the circadian clock oscillator with the external modification oscillator. The implementation of this type of techniques in anatomical problems is highlighted, which are rare in the literature. The implementations will be carried out through different simulations using numerical techniques and the way in which we will determine the coupling conditions of an internal cycle of the system versus external cycles will be detailed. In the final development of this work, we will be able to see in the model without an external modification signal the existence of stable limit cycles and discover the moment in which the synchronization of the internal oscillator and the external modification signal occurs. These types of problems are common when making biological models that are described by a physical analysis.

Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. In addition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given.

Bifurcations of Invariant Torus and Knotted Periodic Orbits in Generalized Hopf-Langford Type Equations

In this paper, we study the bifurcations of invariant torus and knotted periodic orbits for generalized Hopf-Langford type equations. By using bifurcation theory of dynamical systems, we obtain the exact explicit form of the heteroclinic orbits and knot periodic orbits. Moreover, under small perturbation, we prove that the perturbed planar system has two symmetric stable limit cycles created by Poincare bifurcations. Therefore, the corresponding three-dimensional perturbed system has an attractive invariant rotation torus.

Physical reservoir computing with origami and its application to robotic crawling

A new paradigm called physical reservoir computing has recently emerged, where the nonlinear dynamics of high-dimensional and fixed physical systems are harnessed as a computational resource to achieve complex tasks. Via extensive simulations based on a dynamic truss-frame model, this study shows that an origami structure can perform as a dynamic reservoir with sufficient computing power to emulate high-order nonlinear systems, generate stable limit cycles, and modulate outputs according to dynamic inputs. This study also uncovers the linkages between the origami reservoir’s physical designs and its computing power, offering a guideline to optimize the computing performance. Comprehensive parametric studies show that selecting optimal feedback crease distribution and fine-tuning the underlying origami folding designs are the most effective approach to improve computing performance. Furthermore, this study shows how origami’s physical reservoir computing power can apply to soft robotic control problems by a case study of earthworm-like peristaltic crawling without traditional controllers. These results can pave the way for origami-based robots with embodied mechanical intelligence.

Non-smooth Dynamics Emerging from Predator-driven Discontinuous Prey Dispersal

In ecology, the refuge protection of the prey plays a significant role in the dynamics of the interactions between prey and predator. In this paper, we investigate the dynamics of a non-smooth prey-predator mathematical model characterized by density-dependent intermittent refuge protection of the prey. The model assumes the population density of the predator as an index for the prey to decide on when to avail or discontinue refuge protection, representing the level of apprehension of the prey by the predators. We apply Filippov's regularization approach to study the model and obtain the sliding segment of the system. We obtain the criterion for the existence of the regular or virtual equilibria, boundary equilibrium, tangent points, and pseudo-equilibria of the Filippov system. The conditions for the visibility (or invisibility) of the tangent points are derived. We investigate the regular or virtual equilibrium bifurcation, boundary-node bifurcation and pseudo-saddle-node bifurcation. Further, we examine the effects of dispersal delay on the Filippov system associated with prey vigilance in identifying the predator population density. We observe that the hysteresis in the Filippov system produces stable limit cycles around the predator population density threshold in some bounded region in the phase plane. Moreover, we find that the level of apprehension and vigilance of the prey play a significant role in their refuge-dispersion dynamics.

Coexistence of chaotic attractor and unstable limit cycles in a 3D dynamical system

The coexistence of stable limit cycles and chaotic attractors has already been observed in some 3D dynamical systems. In this paper we show, using the T-system, that unstable limit cycles and chaotic attractors can also coexist. Moreover, by completing the characterization of the existence of invariant algebraic surfaces and their associated global dynamics, we give a better understanding on the disappearance of the strange attractor and the limit cycles of the studied system.