Preservation of a two-wing Lorenz-like attractor with stable equilibria

"In this paper, we present the preservation of a two-wing Lorenz-like attractor when in the Lorenz system a feedback control is applied, making two of its equilibria a sink. The forced system is capable of generating bistability and the trajectory settles down at one stable equilibrium point depending on the initial condition when the forced signal is zero. Due to a variation in the coupling strength of the control signal the symmetric equilibria of the Lorenz system move causing the basins of attraction to be the dynamic bounded regions that change accordingly. Thus, the preservation of a two-wing Lorenz-like attractor is possible using a switched control law between these dynamic basins of attraction. The forced switched systems also preserve multistability regarding the coupling strength and present multivalued synchronization according to the basin of attraction in which they were initialized. Bifurcations of the controlled system are used to exemplify the different basins generated by the forcing. An illustrative example is given to demonstrate the approach proposed.

Generation of multiscroll attractors by controlling the equilibria

Investigació

Analog electronic implementation of a class of hybrid dissipative dynamical system

"An analog electronic implementation by means of operational amplifiers of a class of hybrid dissipative systems in R3R3 is presented. The switching systems have two unstable hyperbolic focus-saddle equilibria with the same stability index, a positive real eigenvalue and a pair of complex conjugated eigenvalues with negative real part. The analog circuit generates signals that oscillate in an attractor located between the two unstable equilibria, and may present saturation states at the moment of energizing it, i.e. if the initial voltage on the capacitors do not belong to the basin of attraction the circuit will end on a saturation state.

A family of hyperchaotic multi-scroll attractors in Rn

Investigació

Generalized multistable structure via chaotic synchronization and preservation of scrolls

Investigació

Multivalued synchronization by Poincar coupling

"This work presents multivalued chaotic synchronization via coupling based on the Poincaré plane. The coupling is carried out by an underdamped signal, triggered every crossing event of the trajectory of the master system through a previously defined Poincaré plane. A master–slave system is explored, and the synchronization between the systems is detected via the auxiliary system approach and the maximum conditional Lyapunov exponent. Due to the response to specific conditions two phenomena may be obtained: univalued and multivalued synchronization. Since the Lyapunov exponent is not enough to detect these two phenomena, the distance between the pieces of trajectories of the slave and auxiliary systems with different initial conditions is also used as a tool for the detection of multivalued synchronization. Computer simulations using the benchmark chaotic systems of Lorenz and Rössler are used to exemplify the approach proposed.

Mathematical modelling of the waning of anti-RBD IgG SARS-CoV-2 antibody titers after a two-dose BNT162b2 mRNA vaccination

BackgroundAfter exposure to SARS-CoV-2 and/or vaccination there is an increase in serum antibody titers followed by a non-linear waning. Our aim was to find out if this waning of antibody titers would fit to a mathematical model.MethodsWe analyzed anti-RBD (receptor binding domain) IgG antibody titers and the breakthrough infections over a ten-month period following the second dose of the mRNA BNT162b2 (Pfizer-BioNtech.) vaccine, in a cohort of 54 health-care workers (HCWs) who were either never infected with SARS-CoV-2 (naïve, nHCW group, n=27) or previously infected with the virus (experienced, eHCW group, n=27). Two mathematical models, exponential and power law, were used to quantify antibody waning kinetics, and we compared the relative quality of the goodness of fit to the data between both models was compared using the Akaik Information Criterion.ResultsWe found that the waning slopes were significantly more pronounced for the naïve when compared to the experienced HCWs in exponential (p-value: 1.801E-9) and power law (p-value: 9.399E-13) models. The waning of anti-RBD IgG antibody levels fitted significantly to both exponential (average-R2: 0.957 for nHCW and 0.954 for eHCW) and power law (average-R2: 0.991 for nHCW and 0.988 for eHCW) models, with a better fit to the power law model. In the nHCW group, titers would descend below an arbitrary 1000-units threshold at a median of 210.6 days (IQ range: 74.2). For the eHCW group, the same risk threshold would be reached at 440.0 days (IQ range: 135.2) post-vaccination.ConclusionTwo parsimonious models can explain the anti-RBD IgG antibody titer waning after vaccination. Regardless of the model used, eHCWs have lower waning slopes and longer persistence of antibody titers than nHCWs. Consequently, personalized vaccination booster schedules should be implemented according to the individual persistence of antibody levels

Widening of the basins of attraction of a multistable switching dynamical system with the location of symmetric equilibria

"A switching dynamical system by means of piecewise linear systems in that presents multistability is presented. The flow of the system displays multi-scroll attractors due to the unstable hyperbolic focus-saddle equilibria with stability index of type I, i.e., a negative real eigenvalue and a pair of complex conjugated eigenvalues with positive real part. This class of systems is constructed by a discrete control mode changing the equilibrium point regarding the location of their states. The scrolls appear when the stable and unstable eigenspaces of each adjacent equilibrium point generate the stretching and folding mechanisms needed in chaos, i.e., the unstable manifold in the first subsystem carries the trajectory towards the stable manifold of the immediate adjacent subsystem. The resulting attractors are located around four focus saddle equilibria. If the equilibria are located symmetrically to one of the axes and the distance between each equilibria is properly adjusted to generate two double-scroll chaotic attractors, the system can present from bistable to multistable parallel solutions regarding the position of their initial states. In addition the resulting basin of attraction presents a significatively widening when the distance between the equilibria of the parallel attractors is displaced.

Hyperchaotic encryption based on multi-scroll piecewise linear systems

"A hyperchaotic multi-scroll piecewise linear system in R4 is binarized to generate a pseudo-random sequence which encrypt a grayscale image via symmetric-key algorithm. The sequence is analyzed throughout statistical tests according to the National Institute of Standards and Technology (NIST) specifications. The scrolls of the system are the result of a switching law that changes between the saddle hyperbolic equilibria of piecewise linear systems with eigenvalues as follows: two negative real and one pair of complex conjugate eigenvalues with positive real part. Thus, the encryption quality is evaluated depending on the variation of the number of scrolls.

Generalized multistable structure via chaotic synchronization and preservation of scrolls

"Switched systems are capable of generating chaotic multi-scroll behavior in by means of a control signal. This signal regulates an equilibrium position of the system and is defined according to the number of scrolls that is displayed by the attractor. Thus, if two systems are controlled by different signals, they exhibit a different number of scrolls. Multistability can be created by a pair of unidirectionally coupled unstable dissipative switched linear systems. A theoretical study of this phenomenon is performed with the jerky equations. Generalized synchronization is observed in numerical simulations of the master-salve system with different control signals. The proposed configuration preserves the number of scrolls and can possess an arbitrary large number of coexisting chaotic multi-scroll attractors.