N-dimensional Dynamical System Research Articles

Second-Order Modified Nonstandard Explicit Euler and Explicit Runge–Kutta Methods for n-Dimensional Autonomous Differential Equations

Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems.

Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

For three-dimensional vector fields, the governing formula of invariant manifolds grown from a hyperbolic cycle is given in cylindrical coordinates. The initial growth directions depend on the Jacobians of Poincaré map on that cycle, for which an evolution formula is deduced to reveal the relationship among Jacobians of different Poincaré sections. The evolution formula also applies to cycles in arbitrary finite n-dimensional autonomous continuous-time dynamical systems. Non-Möbiusian/Möbiusian saddle cycles and a dummy X-cycle are constructed analytically as demonstration. A real-world numeric example of analyzing a magnetic field timeslice on EAST is presented.

Fictitious time integration method for seeking periodic orbits of nonlinear dynamical systems

In the paper, we compute the period and periodic orbit of an n-dimensional nonlinear dynamical system by developing two iterative algorithms based on the fictitious time integration method (FTIM). Periodicity condition, also known as the Poincaré map, is a necessary condition for the existence of a periodic motion in the state space, which consists of n implicit nonlinear algebraic equations (NAEs). Instead of solving the NAEs by the Newton–Raphson method, we derive an equivalent nonlinear scalar equation to determine unknown period by using the FTIM. The resulting sequence of the iterated periods monotonically converges to a desired period, wherein the unknown initial point on the periodic orbit is determined simultaneously. We find that the second iterative algorithm using the matrix shape function to convert the periodic problem into a corresponding initial value problem is convergent faster than the first iterative algorithm. A key point is that the periodicity condition is satisfied automatically by the second iterative algorithm when the terminal values of the new variables are convergent. Numerical examples exhibit some major advantages of these two iterative algorithms.

Periodic solutions of nonlinear ordinary differential equations computed by a boundary shape function method and a generalized derivative-free Newton method

In the paper, the period of an n-dimensional nonlinear dynamical system is computed by a formula derived in an (n+1)-dimensional augmented state space. The periodic conditions and nonlinear first-order ordinary differential equations constitute a specific periodic boundary value problem within a time interval, whose length is an unknown finite constant. Two periodic problems are considered: (I) boundary values are given and (II) boundary values are unknown. A boundary shape function method (BSFM), using the derived shape functions, is devised to an initial value problem with the initial values of new variables given, whereas the terminal values and period are determined by iterative algorithms. The periodic solutions obtained by the BSFM satisfy the periodic conditions automatically. For the sake of comparison, the iterative algorithms based on the shooting method are developed, directly implementing the Poincaré map with the fictitious time integration method to determine the periodic solutions, where the periodic conditions are transformed to a mathematically equivalent scalar equation. Owing to the implicit, non-differentiable and nonlinear property of the scalar equation, we develop a generalized derivative-free Newton method (GDFNM) to solve the periodic problem of case (I), which can pick up very accurate period through a few iterations. In numerical examples the computed order of convergence displays the merit of the proposed iterative algorithms. The BSFM and GDFNM are better than the shooting method from the aspects of convergence speed, accuracy and stability. A conventional Poincaré mapping method is introduced to solve the periodic problems with the same parameters. The BSFM converges faster and more accurate than the Poincaré mapping method and is less sensitive to the initial guesses of initial values and period.

Periodic Orbits of Nonlinear Ordinary Differential Equations Computed by a Boundary Shape Function Method

In the paper, we determine the period of an n-dimensional nonlinear dynamical system by using a derived formula in an (n + 1)-dimensional augmented space. To form a periodic motion, the periodic conditions in the state space and nonlinear first-order differential equations constitute a special periodic problem within a time interval with an unknown length. Two periodic problems are considered: (a) boundary values are given and (b) boundary values are unknown. By using the shape functions, a boundary shape function method (BSFM) is devised to obtain an initial value problem with the initial values of the new variables given. The unknown terminal values of the new variables and period are determined by two iterative algorithms for the case (a) and one iterative algorithm for the case (b). The periodic solutions obtained from the BSFM satisfy the periodic conditions automatically. For the numerical example, the computed order of convergence displays the merit of the BSFM. For the sake of comparison, the iterative algorithms based on the shooting method for cases (a) and (b) were developed by directly implementing the Poincaré map into the fictitious time-integration method to determine the period. The BSFM is better than the shooting method in terms of convergence speed, accuracy, and stability.

Impact of Substitution Rate on Energy Consumption Structure: A Dynamical System Approach

Energy structure in global terminal consumption is moving towards a direction of low carbon. This study analyzes the evolution of the share of coal in energy consumption structure by continuous dynamical systems. Based on the complex and dynamic interactions among energy sources, an n-dimensional dynamical system model is constructed where the alternative of other energy sources for coal are modeled by the linear parameters. A case study of China is performed on its terminal energy consumption structure. Three scenarios are set to analyze and compare the particular evolution paths of coal share: the substitution rate changes, the self-growth rate changes, and both the substitution rate and self-growth rate change. Results show that the improvement of renewable power in either the substitution rate or self-growth rate is the most effective to reduce the proportion of coal. As to other energy sources, raising the coal substitution rate is better than increasing their self-growth rate. It concludes that proper control of coal power, moderate development of hydropower, and active and safe development of nuclear power are likely increase the proportion of electric in terminal energy consumption. This study provides an alternative and promising approach to analyze the evolution of coal ratio in energy consumption structure and may have potential application in other areas and countries.

Dual Compound–Compound Synchronization of Twelve n-Dimensional Dynamical Systems

A dual synchronization scheme combined with compound–compound synchronization is proposed. In usual compound–compound synchronization scheme, synchronization has been achieved for six dynamical systems, while in the presented scheme compound–compound synchronization has been developed in the dual synchronization frame involving twelve dynamical systems. Numerous synchronization schemes such as dual compound, dual combination, compound–compound and compound synchronization may be derived as particular cases of the presented scheme. By employing nonlinear control method and Lyapunov stability criteria, controllers are designed. Illustrations are presented using Genesio-Tesi and Wang chaotic systems. Computational results are presented to show the effectiveness of the proposed methodology. A good agreement has been obtained between analytical methodology and numerical treatment.

A Hybrid Control Algorithm for Gradient-Free Optimization using Conjugate Directions

The problem of steering a particular class of n-dimensional continuous-time dynamical systems towards the minima of a function without gradient information is considered. We propose a hybrid controller, implementing a discrete-time Direct Search algorithm based on conjugate directions, able to solve the optimization problem for the resulting closed loop system in an almost global sense. Moreover, we propose a modified version by imposing a lower bound on the step size and able to achieve robust practical convergence to the optimum.

Addendum in ‘Silva, Vin´ıcius B. et al., 2018, A new application of the normal form description to a N-dimensional dynamical systems attending the conditions of a Hopf bifurcation, Journal of Vibration Testing and System Dynamics, 2(3), 249-256’

Addendum in ‘Silva, Vin´ıcius B. et al., 2018, A new application of the normal form description to a N-dimensional dynamical systems attending the conditions of a Hopf bifurcation, Journal of Vibration Testing and System Dynamics, 2(3), 249-256’

Global stabilization of a genetic positive feedback loop via the design of a synthetic auto-repression

Genetic positive feedback loops are essential for cell differentiation processes. They are accurately modeled with N-dimensional non-linear monotone dynamical systems that display bi-stability: the two stable fixed points represent two distinct cell differentiated states, whereas the unstable fixed point is interpreted as a cell undifferentiated state. This paper shows that the synthetic design of a simple self-inhibition of one gene in the loop is able to globally stabilize the unstable fixed point of the network. This modification may lead to a promising cell dedifferentiation process during which cells regress from a specialized state to an earlier developmental state. Compared to a similar experiment designed for the Toggle Switch, this new synthetic circuit prevents the use of any input and measurement devices, reducing greatly the complexity of the biological set-up. In order to take into account inherent biological uncertainties, the cell undifferentiated state is later considered as a region of the state space around the unstable fixed point and is shown to be globally attractive with the same simple synthetic modification of the loop. Some conditions are given such that all the possible fixed points of the circuit are confined in the undifferentiated region and the global results are proved with the theory of monotone dynamical systems.

The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space

We consider the evolution of multi-pulse patterns in an extended Klausmeier equation with parameters that change in time and/or space. We formally show that the full PDE dynamics of a N-pulse configuration can be reduced to a N-dimensional dynamical system describing the dynamics on a N-dimensional manifold MN. Next, we determine the local stability of MN via the quasi-steady spectrum associated to evolving N-pulse patterns, which provides explicit information on the boundary ∂MN. Following the dynamics on MN, a N-pulse pattern may move through ∂MN and ‘fall off’ MN. A direct nonlinear extrapolation of our linear analysis predicts the subsequent fast PDE dynamics as the pattern ‘jumps’ to another invariant manifold MM, and specifically predicts the number N−M of pulses that disappear. Combining the asymptotic analysis with numerical simulations of the dynamics on the various invariant manifolds yields a hybrid asymptotic–numerical method describing the full process that starts with a N-pulse pattern and typically ends in the trivial homogeneous state without pulses. We extensively test this method against PDE simulations and deduce general conjectures on the nature of pulse interactions with disappearing pulses. We especially consider the differences between the evolution of irregular and regular patterns. In the former case, the disappearing process is gradual: irregular patterns lose their pulses one by one. In contrast, regular, spatially periodic, patterns undergo catastrophic transitions in which either half or all pulses disappear. However, making a precise distinction between these two drastically different processes is quite subtle, since irregular N-pulse patterns that do not cross ∂MN typically evolve towards regularity. hybrid asymptotic-numerical method

A new application of the normal form description to a N-dimensional dynamical systems attending the conditions of a Hopf bifurcation

In this paper we prove the following theorem: consider a N-dimensional dynamical system that is reduced to its center manifold. If it is proved the system satisfies the conditions of a Hopf bifurcation theorem then the original system of differential equations escribing the dynamics can be rewritten in a simpler analytical expression that preserves the phase space topology. The theorem proposed and proven effectively reduces the work done to obtain the normal form for the class of dynamical systems with the ccurrence of Hopf bifurcation.

On novel conditions of chaotic attractors existence in autonomous polynomial dynamical systems

The novel sufficient conditions for the existence of chaotic dynamics in n-dimensional autonomous polynomial dynamical systems are found. These conditions allowed to extend collections of known systems, for which the existence of chaotic attractors is rigorously proved and to complement these collections by new chaotic systems. As an example, a new chaotic attractor generated by the known Rabinovich–Fabrikant system is shown. In addition, new discrete maps generating chaos in Rabinovich–Fabrikant system are also represented. Besides, an application of got results for research of cubic dynamical systems is demonstrated.

Fundamental aspects of curvature indices for characterizing dynamical systems

It is important to characterize the properties of dynamical systems by a quantity that signifies their structural changes, in particular those associated with occurrence of chaos or other transitional behaviors. There are some well-known indices, such as Lyapunov exponent, fractal dimension, and Kolmogorov entropy, while in this article we use a new quantifier, named the curvature index, to study the dynamical systems. The curvature index (proposed by Chen and Chang in Chaos 22(2):371–383 2012) is defined as the limit of the average curvature of a trajectory during evolution for a dynamical system, which lumps all the bending effects of the trajectory to a number, and estimates its average size (such as an attractor) in virtue of an inscribed space ball. One may define N-1 curvature indices for an N-dimensional dynamical system. Once the system undergoes a structural change, there are corresponding changes in the first and/or higher curvatures. The study is aimed to examine fundamental aspects of the curvature indices with further applications to some outstanding examples of dynamical systems in the literature, in parallel to the analysis by the Lyapunov exponents.

Geometrical and spectral properties of Pisot substitutions

A necessary condition for an n-dimensional substitutive dynamical system to be a finite extension of an algebraic system is that the substitution be Pisot. We discuss the sufficiency of the Pisot property and review conditions under which such an extension is guaranteed to be almost everywhere one-to-one.

Generalization of combination–combination synchronization of chaotic n-dimensional fractional-order dynamical systems

The generalization of combination–combination (C–C) synchronization of chaotic n-dimensional (nD) fractional-order \((0<\alpha \le 1)\) dynamical systems is studied. Firstly, we replace arbitrary four chaotic nD ordinary dynamical systems by four chaotic nD fractional-order dynamical systems which have unique solutions. Secondly, we extend the scheme of a recent paper (Sun et al. in Nonlinear Dyn 73: 1211–1222, 2013) to study the generalization of C–C synchronization among four nD fractional-order dynamical systems. Examples of combination–combination synchronization among four identical or different of 6D chaotic fractional-order systems are discussed. The analytical formula of the control functions is tested numerically to achieve C–C synchronization, and good agreement is found.

Projective Synchronization of Chaotic Discrete Dynamical Systems via Linear State Error Feedback Control

A projective synchronization scheme for a kind of n-dimensional discrete dynamical system is proposed by means of a linear feedback control technique. The scheme consists of master and slave discrete dynamical systems coupled by linear state error variables. A kind of novel 3-D chaotic discrete system is constructed, to which the test for chaos is applied. By using the stability principles of an upper or lower triangular matrix, two controllers for achieving projective synchronization are designed and illustrated with the novel systems. Lastly some numerical simulations are employed to validate the effectiveness of the proposed projective synchronization scheme.

Properties of networks with partially structured and partially random connectivity.

Networks studied in many disciplines, including neuroscience and mathematical biology, have connectivity that may be stochastic about some underlying mean connectivity represented by a non-normal matrix. Furthermore, the stochasticity may not be independent and identically distributed (iid) across elements of the connectivity matrix. More generally, the problem of understanding the behavior of stochastic matrices with nontrivial mean structure and correlations arises in many settings. We address this by characterizing large random N×N matrices of the form A=M+LJR, where M,L, and R are arbitrary deterministic matrices and J is a random matrix of zero-mean iid elements. M can be non-normal, and L and R allow correlations that have separable dependence on row and column indices. We first provide a general formula for the eigenvalue density of A. For A non-normal, the eigenvalues do not suffice to specify the dynamics induced by A, so we also provide general formulas for the transient evolution of the magnitude of activity and frequency power spectrum in an N-dimensional linear dynamical system with a coupling matrix given by A. These quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point. We derive these formulas and work them out analytically for some examples of M,L, and R motivated by neurobiological models. We also argue that the persistence as N→∞ of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of A, as previously observed, arises in regions of the complex plane Ω where there are nonzero singular values of L(-1)(z1-M)R(-1) (for z∈Ω) that vanish as N→∞. When such singular values do not exist and L and R are equal to the identity, there is a correspondence in the normalized Frobenius norm (but not in the operator norm) between the support of the spectrum of A for J of norm σ and the σ pseudospectrum of M.

Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.

The Briot–Bouquet systems and the center families for holomorphic dynamical systems

We give a complete solution to the existence of isochronous center families for holomorphic dynamical systems. The study of center families for n-dimensional holomorphic dynamical systems naturally leads to the study of (n−1)-dimensional Briot–Bouquet systems in the phase space. We first give a detailed study of the Briot–Bouquet systems. Then we show the existence of isochronous center families in the neighborhood of the equilibrium point of three-dimensional systems based on the two-dimensional Briot–Bouquet theory. The same approach works in arbitrary dimensions.