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When two different groups of cells are placed next to each other, they compete with each other to get enough space to increase volume, and nutrients according to the kind and growth conditions. In order to further discuss the competition of normal and cancer cells in the human body, a modified Lotka–Volterra competition model is introduced to develop a symbiosis analysis model of cell groups. Lotka–Volterra equation is one of the mathematical models that is used to study the competition of these cell groups and has been applied to determine the dominant and recessive cell groups. In this work, we have modified this equation to obtain better results of cancer cells competition. Our obtained results show that by using the modified equation, the number of destroyed cells during the competition is also considered.

Nonlinear dynamic systems can be considered in terms of feedback circuits (for short, circuits), which are circular interactions between variables. Each circuit can be identified without ambiguity from the Jacobian matrix of the system. Of special relevance are those circuits (or unions of disjoint circuits) that involve all the variables of the system. We call them "nuclei" because, in the same way as, in Biology, the cell nucleus contains essential genetic information, in nonlinear dynamics, nuclei are crucial elements in the genesis of steady states. Indeed, each nucleus taken alone can generate one or more steady states, whose nature is determined by the sign patterns of the nucleus. There can be up to two nuclei in 2D systems, six in 3D systems, … n! in nD systems. However, many interesting systems of high dimensionality have only two, sometimes even one, nucleus.

This paper is based on an extensive exploitation of the Jacobian matrix of systems, in order to figure the global structure of phase space. In nonlinear systems, the value, and often the sign, of terms of the Jacobian matrix depend on the location in phase space. In contrast with a current usage, we consider this matrix, as well as its eigenvalues and eigenvectors, not only in close vicinity of steady states of the system, but also everywhere in phase space. Two distinct, but complementary approaches are used here.

In Sec. 2, we define frontiers that partition phase space according to the signs of the eigenvalues (or of their real part if they are complex), and where required, to the slopes of the eigenvectors. From then on, the exact nature of any steady state that might be present in a domain can be identified on the sole basis of its location in that domain and, in addition, one has at least an idea of the possible number of steady states.

In Sec. 3, we use a more qualitative approach based on the theory of circuits. Here, phase space is partitioned according to the sign patterns of elements of the Jacobian matrix, more specifically, of the nuclei. We feel that this approach is more generic than that described in Sec. 2. Indeed, it provides a global view of the structure of phase space, and thus permits to infer much of the dynamics of the system by a simple analysis of sign patterns within the Jacobian matrix. This approach also turned out to be extremely useful for synthesizing systems with preconceived properties.

For a wide variety of systems, once the partition process has been achieved, each domain comprises at most one steady state. We found, however, a family of systems in which two steady states (typically, two stable or two unstable nodes) differ by neither of the criteria we use, and are thus not separated by our partition processes. This counter-example implies non-polynomial functions.

It is essential to realize that the frontier diagrams permit a reduction of the dimensionality of the analysis, because only those variables that are involved in nonlinearities are relevant for the partition process. Whatever the number of variables of a system, its frontier diagram can be drawn in two dimensions whenever no more than two variables are involved in nonlinearities.

Pre-existing conjectures concerning the necessary conditions for multistationarity are discussed in terms of the partition processes.

A neuro-fuzzy based model is proposed in this paper for estimating the Lyapunov exponents of an unknown dynamical system according solely to a set of observations. Several approaches have been presented in recent years; most of them using the approximation of both the function of the trajectory of observations and the partial derivatives, to yield the Jacobian matrix of the function. The Jacobian matrix is then employed in the Jacobian-based methods that extract the Lyapunov exponents by QR-decomposition. While the accurate estimation of Lyapunov exponents has been sought, most of the related papers mainly focus on the accuracy of the trajectory approximation. In this paper, an Adaptive Neuro-Fuzzy Inference System is presented and stated to be an efficient tool for such a purpose. Structural parameters of the proposed model as the embedding dimension and the delay time are calculated by the Takens theorem and autocorrelation function, respectively. Model validation is performed by cross approximate entropy. Then, the promising performance of the proposed model as an accurate estimation of the Lyapunov exponents and its robustness to the measurement noise are finally evaluated.

The Cardiac Purkinje Fiber (CPF) is the last branch of the heart conduction system, which is meshed with the normal ventricular myocyte. Purkinje fiber plays a key role in the occurrence of ventricular arrhythmia and maintenance. Does the heart Purkinje fiber cells have the same memory function as the cerebral nerve? In this paper, the cardiac Hodgkin–Huxley equation is taken as the object of study. In particular, we find that the potassium ion-channel $K$ and the sodium ion-channel $Na$ are memristors. We also derive the small-signal equivalent circuits about the equilibrium points of the CPF Hodgkin–Huxley model. According to the principle of local activity, the regions of Locally-Active domain, Edge of Chaos domain and Locally-Passive domain are partitioned under parameters $(a,b)$, and the domain exhibiting the normal human heartbeat frequency range (Goldilocks Zone) is identified. Meanwhile, the Super-Critical Hopf bifurcation of the CPF Hodgkin–Huxley model is identified. Finally, the migration changes between different state domains under external current $I_{\mathrm{\text{ext}}}$ excitation are analyzed in detail.

All of the above complex nonlinear dynamics are distilled and mapped geometrically into a surreal union of intersecting two-dimensional manifolds, dubbed the Hodgkin–Huxley’smagic roof.

The construction of multidimensional discrete hyperchaotic maps with ergodicity and larger Lyapunov exponents is desired in cryptography. Here, we have designed a general $n$D ($n\ge 2$) discrete hyperchaotic map ($n$D-DHCM) model that can generate any nondegenerate $n$D chaotic map with Lyapunov exponents of desired size through setting the control matrix. To verify the effectiveness of the $n$D-DHCM, we have provided two illustrative examples and analyzed their dynamic behavior, and the results demonstrated that their state points have ergodicity within a sufficiently large interval. Furthermore, to address the finite precision effect of the simulation platform, we analyzed the relationship between the size of Lyapunov exponent and the randomness of the corresponding state time sequence of the $n$D-DHCM. Finally, we designed a keyed parallel hash function based on a 6D-DHCM to evaluate the practicability of the $n$D-DHCM. Experimental results have demonstrated that $n$D discrete chaotic maps constructed using $n$D-DHCM have desirable Lyapunov exponents, and can be applied to practical applications.

We would like to show that complex dynamics can be deciphered and at least partly understood in terms of its underlying logical structure, more specifically in terms of the feedback circuits built in the differential equations. This approach has permitted to build a number of new three- and four-variable systems displaying chaotic dynamics.

Starting from the well-known Rössler equations for deterministic chaos, one asks how systems with the same types of steady states can be synthesized ab initio from appropriate feedback circuits (= appropriate logical structure). It is found that, granted an appropriate logical structure, the existence of a domain of chaotic dynamics is remarkably robust towards changes in the nature of the nonlinearity used, and towards those sign changes which respect the nature (positive vs negative) of the feedback circuits. Using logical arguments, it was also easy to find related systems with a single steady state. A variety of 3- and 4-d systems based on other combinations of feedback circuits and generating chaotic dynamics are described.

The aim of this work is to contribute to a better understanding of the respective roles of feedback circuits and nonlinearity — both essential — in so-called "non trivial behavior", including deterministic chaos. Special emphasis is put on the interest of using the Jacobian matrix and its by-products (characteristic equation, eigenvalues, …) not only close to steady states (where linear stability analysis can be performed) but also elsewhere in phase space, where precious indications about the global behavior can be collected.

Some types of rigid origami possess specific geometric properties. They have a single degree of freedom, and can experience large configuration changes without cut or being stretched. This study presents a numerical analysis and finite element simulation on the folding behavior of deployable origami structures. Equivalent pin-jointed structures were established, and a Jacobian matrix was formed to constrain the internal mechanisms in each rigid plane. A nonlinear iterative algorithm was formulated for predicting the folding behavior. The augmented compatibility matrix was updated at each step for correcting the incompatible strains. Subsequently, finite element simulations on the deployable origami structures were carried out. Specifically, two types of generalized deployable origami structures combined by basic parts were studied, with some key parameters considered. It is concluded that, compared with the theoretical values, both the solutions obtained by the nonlinear algorithm and finite element analysis are in good agreement, the proposed method can well predict the folding behavior of the origami structures, and the error of the numerical results increases with the increase of the primary angle.

Let L be a finitely generated free color Lie superalgebra. We obtained a criterion in terms of invertibility of the double Jacobian matrix associated with a special element for a given endomorphism of L to be an automorphism.

In this note, we investigate Jacobian conjecture through investigation of automorphisms of polynomial rings in characteristic $p$. Making use of the technique of inverse limits, we show that under Jacobian condition for a given homomorphism $\phi$ of the polynomial ring k[$x_1,\dots ,x_n$], if $\phi$ preserves the maximal ideals, then $\phi$ is an automorphism.

A systematic methodology for solving the inverse dynamics of a 6-PRRS parallel robot is presented. Based on the principle of virtual work and the Lagrange approach, a methodology for deriving the dynamical equations of motion is developed. To resolve the inconsistency between complications of established dynamic model and real-time control, a simplifying strategy of the dynamic model is presented. The dynamic character of the 6-PRRS parallel robot is analyzed by example calculation, and a full and precise dynamic model using simulation software is established. Verification results show the validity of the presented algorithm, and the simplifying strategies are practical and efficient.

This paper proposes a method calculating joint velocities of a robot which moves the end effector at desired velocity where some of the joint motions are constrained. It is an extension of the Resolved Motion Rate Control (RMRC) method which has been used in cases where there is no constraint on the motion of the joints. The proposed method is called the extended RMRC (E-RMRC). Though the E-RMRC is expressed in a simple form, application of the E-RMRC to a specific robot system is not straightforward and sometimes calls for elaboration. So, the paper describes the application of the E-RMRC to the motion of a mobile manipulator. The example explains how the proposed method is applied to find the joint rate to move the end effector of the mobile manipulator through a desired trajectory while the trajectory of the mobile base is constrained. The application is tested and verified through simulation and experiments.

The relationship of velocity between the end of the manipulator and each joint is discussed in the basis coordinate system. The inverse velocity is analyzed using Jacobian matrix method for the manipulator. Considering its practical geometric parameters, restriction and physical characteristic, the virtual prototype of the robot is established in ADAMS, and then the co-simulation was completed by applying the ADAMS/Controls and MATLAB/Simulink. The virtual prototyping model of the robot provides a basis for research on off-line programming of the modular robot.