Narrow Results

Ultimate boundedness of chaotic dynamical systems is one of the fundamental concepts in dynamical systems, which plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors and the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, chaos synchronization. However, it is often difficult to obtain the bounds of the hyperchaotic systems due to the complex algebraic structure of the hyperchaotic systems. This paper has investigated the boundedness of solutions of a nonlinear hyperchaotic system. We have obtained the global exponential attractive set and the ultimate bound set for this system. To obtain the ellipsoidal ultimate bound, the ultimate bound of the proposed system is theoretically estimated using Lagrange multiplier method, Lyapunov stability theory and optimization theory. To show the ultimate bound region, numerical simulations are provided.

The article is devoted to the study of the global behavior of the generalized Rabinovich system describing the process of interaction between waves in plasma. For this generalized system, we obtain the positive invariant set (ultimate bound) and globally exponential attractive set using the approach where we transform the initial problem of finding the corresponding set to the conditional extremum problem and solve this problem. Furthermore, the rate of the trajectories going from the exterior of the attractive set to the interior of the attractive set is also obtained. Numerical localization of attractor is presented. Meanwhile, the volumes of the ultimate bound set and the global exponential attractive set are obtained, respectively. The main innovation of this article lays in considering the generalized form of the Rabinovich system with $v_3>0$ and obtaining the results on the ultimate bound set and globally exponential attractive set for this general case.

We analyze a mathematical model for the treatment of chronic myeloid leukemia (CML). The model is designed for complete recovery of CML patients after treatment. The model developed in the paper [Bunimovich-Mendrazitsky S, Kronik N, Vainstein V, Optimization of interferon-alpha and imatinib combination therapy for CML: A modeling approach, Adv Theory Simul 2(1):1800081, 2018] introduced a combined treatment of CML based on imatinib therapy and immunotherapy. Immunotherapy based on Interferon alpha-2a (IFN-$\alpha$) affects stem and mature cancer cell mortality, and leads to outcome improvements in the combined therapy. The qualitative character of our results shows that additional therapy for the complete cure of CML patients is required. This additional treatment is tumor infiltrating lymphocytes (TIL) along with a combination imatinib and IFN-$\alpha$ treatment. The model examines the interaction between CML cancer cells and effector cells, using an ODE system. Stability analysis of the model defines conditions when imatinib treatment might lead to the eradication of CML with IFN-$\alpha$ and TIL. Three equilibria are investigated for the proposed model. Stability conditions for equilibria are formulated in terms of the linear matrix inequalities (LMIs).

Answering one of the main questions of [S.-D. Friedman, T. Hyttinen and V. Kulikov, Generalized descriptive set theory and classification theory, Mem. Amer. Math. Soc.230(1081) (2014) 80, Chap. 7], we show that there is a tight connection between the depth of a classifiable shallow theory $T$ and the Borel rank of the isomorphism relation ${\cong }_T^{\kappa }$ on its models of size $\kappa$, for $\kappa$ any cardinal satisfying ${\kappa }^{<\kappa }=\kappa >2^{{\aleph }_0}$. This is achieved by establishing a link between said rank and the ${{ℒ}}_{\infty \kappa }$-Scott height of the $\kappa$-sized models of $T$, and yields to the following descriptive set-theoretical analog of Shelah’s Main Gap Theorem: Given a countable complete first-order theory $T$, either ${\cong }_T^{\kappa }$ is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is ${\kappa }^{+}>{\aleph }_1$), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah’s theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of ${\cong }_T^{\kappa }$, and provide a characterization of categoricity of $T$ in terms of the descriptive set-theoretical complexity of ${\cong }_T^{\kappa }$.

In 2012, the scientific world has lost N.N. Krasovskii, an extraordinary mathematician and mechanician whose works have prominently influenced a number of mathematic fields: differential equations, optimal control, and dynamic games theory. His research had profound effects on the development of mathematical research and education. In addition to his distinguished scientific contributions many of N.N. Krasovskiis disciples are academicians and corresponding members of the Russian Academy of Sciences, doctors and candidates of science, professors and engineers.

Consider a C¹ vector field X on a finite-dimensional manifold M and x_e an equilibrium point for the dynamics . We will prove that if I is a C² constant of motion for X such that x_e is also a critical point of I, then is a constant of motion for the linearized system . As a consequence we will give an instability result under the condition that is an indefinite quadratic form. If x_e is not a critical point of I, then we obtain as a constant of motion for the linearized system.

In this paper we give a method to stabilize asymptotically the Lyapunov stable equilibrium states of a system describing the planar motions of an autonomous underwater vehicle.

In this paper we analyze the quadratic and homogeneous Hamiltonian systems on (𝔰𝔬(3))* from the Poisson dynamics and geometry point of view.

In this paper we give a method to stabilize asymptotically the nontrivial Lyapunov stable equilibrium states of the Rabinovich dynamical system.

In this paper, we analyze from the Poisson dynamics and geometry point of view the Ishii dynamical system.

Human immunodeficiency virus (HIV) is a lenti-virus (a member of the retrovirus family) that causes acquired immunodeficiency syndrome (AIDS), a critical condition in humans in which progressive failure of the immune system allows life-threatening opportunistic infections. Over the past few years HIV has been spreading rapidly in the population. Almost, everyday there are thousands of new human cases of HIV infection being recorded in the world and these occur in almost every country of the world. However, the spread of HIV is relatively faster in the developing countries as compared to developed countries because developing countries have limited resources. Worldwide, 70% of HIV infections in the adults have been transmitted through heterosexual contact and vertical transmission accounts for more than 90% of global infection in infants and children. In this paper, we propose a nonlinear mathematical model to study the spread of HIV by considering transmission of disease by heterosexual contact and vertical transmission. A stage structured model is proposed and analyzed by considering the total population variable and dividing the whole population under consideration into three stages: children, adults and old. Also, in this paper it is assumed that the rates of recruitment are different in different groups of population. Equilibria of the model and their stability are also discussed. Using the stability theory of differential equations and computer simulation, it is shown that due to the increase in the awareness of the disease in the adult class the total infective population decreases in the region under consideration.

In this paper, a nonlinear mathematical model is presented for the transmission dynamics of HIV/AIDS in Cuba. Due to Cuba's highly successful national prevention program, we assume that the only mode of transmission is through contact with those yet to be diagnosed with HIV. We find the equilibria of the governing nonlinear system, perform a linear stability analysis, and then provide results on global stability.

This brief article is directed towards young readers, perhaps in their teens, who might be thinking about their future careers and the paths that they might like to follow in later life. I give them for consideration an outline of the fun and excitement that I have myself experienced in scientific research, and encourage them to follow me. To do this, I start by giving some remarks about my own early life and its choices. I then identify some of the decisions and lucky breaks that led to my main scientific work on the theories of stability, nonlinear dynamics and chaos. I point to some new frontiers where knowledge of the physical sciences is spreading into wider areas such as the bio-mechanics of DNA and the prediction of tipping points that could dramatically increase global warming. Finally I give some detailed advice that could be useful as you hopefully enter the thrilling world of scientific research.

In this brief article, I present the surprising, symmetry-breaking behavior of a homogeneous log of maple wood. The log is of square cross-section, and is placed to float in a tank of water. The straight symmetric floating state turns out to be unstable, and it immediately lolls over to an unexpected angle. This could explain why, when I was a young kid, my carefully made boats often did not float upright.

"Visible universe" is a spherical matter crust that rotates at a convenient speed around a central massive body which represents a black-hole. In addition it expands itself in all radial directions. Such structure was first postulated in 2004 and now is fully confirmed by experimental observations from the WMAP (Wilkinson Microwave Anisotropy Probe) released by NASA in March 2007. Using stability theory (ST) we explain present state and future evolution up to final reach of a stable dynamical equilibrium. We present a consistent set of closed form equations that determine basic quantities as radius, age, Hubble constant, mass, density and "missing mass". At the end of the expansion the number of typical stars of visible universe will be equal to the Avogadro Number.

With reference to the restricted two-body problem we show that Stability Theory (ST) and Special Relativity (SR) can be joined together in a new theory that explains a large class of physical phenomena (e.g. black-holes, cosmological dynamics) and overcomes the dualism between SR and General relativity (GR). After recalling the main features of ST (from the Method of Lyapunov to more recent developments up to analysis of fractals) we determine the canonic relativistic equations of the restricted two-body problem. A substantial novelty with respect to noted formulations is pointed out: three state variables (and not two only) are needed for "defining" said equations. They include variable v (magnitude of the rotation speed) in addition to radius and to radial speed. By means of eigenvalue analysis and by application of the Lyapunov theorem on stability in the first approximation we show that linearized system analysis gives a necessary condition only for stability: the radius must be greater than half the Schwarzschild radius. The derivation of a sufficient condition passes through the definition of a convenient Lyapunov function that represents the "local energy" around a given Equilibrium Point. Such derivation is deferred to Part II and results in the proof that the Schwarzschild radius actually represents the reference stable radius of the two-body problem.

We point out a sufficient condition for existence of a stable attractor in the two-body restricted problem. The result is strictly dependent on making reference to relativistic equations and could not be derived from classical analysis. The radius of the stable attractor equals the well known Schwarzschild radius of General Relativity (GR). So we establish a bridge between Special Relativity (SR) and GR via Stability Theory (ST). That opens one way to an innovative study of black-holes and of the cosmological problem. A distinguishing feature is that no singularities come into evidence. The application of the Direct Method of Lyapunov (with a special Lyapunov function that represents the local energy) provides us the theoretical background.

One synchronization method is presented for a fractional-order electronic oscillator only via a scalar signal drive and linear feedback. The synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous.