Chaotic system constructed by product trigonometric function and polynomial and applied to color image encryption

In the applied research of nonlinear system, the low degree of chaos in the dynamical system leads to the limitation of using the chaos method to solve some practical problems. In this paper, we use the product trigonometric function and ternary polynomial to build a dynamical system, which has strong chaotic characteristics. The dynamical system is constructed by two product trigonometric functions and a ternary linear equation, and its chaotic properties are verified by bifurcation diagrams, Lyapunov exponents, fractal dimensions, etc. The system has many parameters and large parameter intervals and is not prone to cycles. The conditions for the non-divergence of this system are given by mathematical derivation, and it is found that the linear part of the system can be replaced by an arbitrary ternary polynomial system and still not diverge, and the bifurcation diagram is drawn to verify it. Finally, the chaotic sequence is distributed more uniformly in the value domain space by adding the modulo operation. Then, the bit matrix of multiple images is directly permuted by the above system, and the experiment confirms that the histogram, information entropy, and pixel correlation of its encrypted images are satisfactory, as well as a very large key space.

The Investigation of Nonlinear Polynomial Control Systems

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

Stability of Non-Hyperbolic Equilibrium Point for Polynomial System of Differential Equations

In this paper, a polynomial system of plane differential equations is observed. The stability of the non-hyperbolic equilibrium point was analyzed using normal forms. The nonlinear part of the differential equation system is simplified to the maximum. Two nonlinear transformations were used to simplify first the quadratic and then the cubic part of the system of equations.

Iterative stability analysis for general polynomial control systems

The main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

Approximate Analytical Solution of a Power-Law Pseudoplastic Fluid in a Boundary Layer over a Porous Plate: A New Application of Multi-Stage Parker-Sochacki Method

In this paper, based on Parker-Sochacki method for solving a system of differential equations,a multistage technique is developed for solving the nonlinear boundary layer equations of powerlawfluid on infinite domain. The problem domain is split into subintervals over which the boundaryvalue problem is replaced with a sequence of subproblems. In a shooting-like approach, the boundarycondition at infinity is converted to an equivalent initial condition. By recasting the problem as apolynomial system of first-order autonomous equations, the sub-problems are solved with Parker-Sochacki method with very high accuracy. The interval of convergence of the solution is deriveda-priorly in terms of the parameters of the polynomial system, which guides optimal choice of thediscretization parameter. The technique yielded a convergent piecewise continuous solution over theproblem domain. The results obtained, demonstrated graphically and in tables, compared well withexisting ones in the literature.

Tripeptide loop closure: a detailed study of reconstructions based on Ramachandran distributions

Tripeptide loop closure (TLC) is a standard procedure to reconstruct protein backbone conformations, by solving a zero dimensional polynomial system yielding up to 16 solutions. In this work, we first show that multiprecision is required in a TLC solver to guarantee the existence and the accuracy of solutions. We then compare solutions yielded by the TLC solver against tripeptides from the Protein Data Bank. We show that these solutions are geometrically diverse (up to 3 Angstroms RMSD with respect to the data), and sound in terms of potential energy. Finally, we compare Ramachandran distributions of data and reconstructions for the three amino acids. The distribution of reconstructions in the second angular space (φ2 , ψ2) stands out, with a rather uniform distribution leaving a central void. We anticipate that these insights, coupled to our robust implementation in the Structural Bioinformatics Library (https://sbl.inria.fr/doc/Tripeptide_loop_closure-user-manual.html), will boost the interest of TLC for structural modeling in general, and the generation of conformations of flexible loops in particular.

Damping parameters indentification of Lienard polynomial system

In many applications of physics, biology, and other sciences, an approach based on the concept of model equations is used as an approximate model of complex nonlinear processes. The basis of this concept is the provision that a small number of characteristic types movements of simple mathematical models inherent in systems give the key to understanding and exploring a huge number of different phenomena. In particular, it is well known that the complex oscillatory motion can be modeled by a system consisting of one or more coupled nonlinear oscillators that governs by differential equation of a second-order. A Lienard system, namely $ \ddot x(t)+f(x(t))\dot x(t)+g(x(t)) = 0$, is a generalization of the such models. Here $f(x(t))$ and $g(x(t))$ are functions that represent various nonlinear phenomena. The typical sources of nonlinearities in Lienard systems are as follows: large displacements of the structure provoking geometric nonlinearities, a nonlinear material behavior, complex law of damping dissipation, etc. In fact, parameter identification is the base of several engineering tasks: identification can be used for the following: (i) to gain knowledge about the process behavior, (ii) to validate theoretical models, (iii) to tune controller parameters, (iv) to design adaptive control algorithms, (v) to process supervision and fault detection, (vi) to on-line optimization. Hence, in order to represent these nonlinearities, identifying the parameters characterizing their behaviors is essential. The problem of constructing globally convergent identificator for polynomial representation of damping force in general Lienar oscillator is addressed. The method of invariant relations is used for identification scheme design. This aproach is based on dynamical extension of original system and construct of appropriate invariant relations, from which the unknowns parameters can be expressed as a functions of the known quantities on the trajectories of extended system. The final synthesis is carried out from the condition of obtaining asymptotic estimates of unknown parameters. It is shown that an asymptotic estimate of the unknown states can be obtained by rendering attractive an appropriately selected invariant manifold in the extended state space.

Combinatorial QFT and the Jacobian Conjecture

The Jacobian Conjecture states that any complex n-dimensional locally invertible polynomial system is globally invertible with polynomial inverse. In 1982, Bass et al. proved an important reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension n. We show in this chapter a result concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter 0≤n′≤n, parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We exhibit a proof, in a QFT formulation, using the intermediate field method exposed in Chapter 3.

Optimization of “human-machine-environment” system: human operator reliability assessment

This article highlights the need to take into account the human component when solving optimization problems in the “man-machine-environment” system. However, currently there are no methods for quantifying the human operator reliability in such systems. An approach for assessing the human operator reliability is proposed. The mathematical apparatus for describing the human operator reliability in the ergatic system based on the Markov theory of random state transitions described by a polynomial system of differential equations for each human operation in the system in a theoretical form is presented. The implementation of this mathematical apparatus assumes the availability of statistical and experimental data on the failure rates and restoration of its productivity. The possibility of the designer to assess the final probability of the human operator of the ergatic environmental control system for one of the most critical operations (to achieve the final goal) in the absence of statistical (experimental) data by analogy with the “weak” link in the technical system is justified. The methodological solution for assessing the human operator reliability in an ergatic control system based on experimental data using the example of a manual trigger of a fire alarm system, based on the expression of a standard pointer in the form of a readiness factor of technical products is shown. The experiment was conducted on a stand assembled from commercially available products, simulating a fire alarm device. In the article the experimental research results by four age groups and their statistical processing are presented.