Exponential Nonlinear Term Research Articles

Extended observer‐based hybrid tracking control strategy for networked system with FDI attacks

AbstractThis paper studies the speed tracking control of networked control systems (NCSs) with external disturbance and false data injection (FDI) attacks. First, the system model with external disturbances and FDI attacks is built. Then, an extended observer based on discrete time sliding function and neural network (NN) is proposed to observe the extended states and suppress the effect of external disturbance and FDI attacks. Furthermore, a novel hybrid discrete‐time sliding mode control (HDSMC) strategy combining discrete time sliding mode control with super‐twisting control is designed to perform closed‐loop control of the system, in which the exponential term and nonlinear term are constructed to restrain the jitters. The convergence and reachability of the sliding motion are proofed. Finally, the validity and feasibility of the proposed methods are proved by simulations and experiments.

A TUTORIAL INTRODUCTION TO THE TWO-SCALE FRACTAL CALCULUS AND ITS APPLICATION TO THE FRACTAL ZHIBER–SHABAT OSCILLATOR

In this paper, a tutorial introduction to the two-scale fractal calculus is given. The two-scale fractal derivative is conformable with the traditional differential derivatives. When the fractal dimensions tend to an integer value, its basic properties are discussed, and the fractal Zhiber–Shabat oscillator is used as an example to reveal the basic properties of a fractal differential equation. The two-scale transform is used to convert the nonlinear Zhiber–Shabat oscillator with the fractal derivatives to the traditional model. The homotopy perturbation method has been demonstrated under a suitable transformation of the system containing several exponential nonlinear terms to the famous Helmholtz–Duffing oscillator. Stability behavior is discussed. Several numerical illustrations are also provided to exhibit the integrity of the introduced formulation. It is demonstrated that the proposed formulation is accurate enough for highly nonlinear differential equations containing large nonlinear terms.

Homotopy Perturbation Method for the Fractal Toda Oscillator

The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and a new form of the Toda oscillator is obtained free of the exponential nonlinear term, which is similar to the form of the Jerk oscillator. The homotopy perturbation method is used to solve the fractal Toda oscillator, and the analytical solution is examined using the numerical solution which shows excellent agreement. Furthermore, the effect of the order of the fractal derivative on the vibration property is elucidated graphically.

Adaptive pseudo-time methods for the Poisson–Boltzmann equation with Eulerian solvent excluded surface

This work further improves the pseudo-transient approach for the Poisson Boltzmann equation (PBE) in the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is known to involve many difficulties, such as exponential nonlinear term, strong singularity by the source terms, and complex dielectric interface. Recently, a pseudo-time ghost-fluid method (GFM) has been developed in [S. Ahmed Ullah and S. Zhao, Applied Mathematics and Computation, 380, 125267, (2020)], by analytically handling both nonlinearity and singular sources. The GFM interface treatment not only captures the discontinuity in the regularized potential and its flux across the molecular surface, but also guarantees the stability and efficiency of the time integration. However, the molecular surface definition based on the MSMS package is known to induce instability in some cases, and a nontrivial Lagrangian-to-Eulerian conversion is indispensable for the GFM finite difference discretization. In this paper, an Eulerian Solvent Excluded Surface (ESES) is implemented to replace the MSMS for defining the dielectric interface. The electrostatic analysis shows that the ESES free energy is more accurate than that of the MSMS, while being free of instability issues. Moreover, this work explores, for the first time in the PBE literature, adaptive time integration techniques for the pseudo-transient simulations. A major finding is that the time increment $\Delta t$ should become smaller as the time increases, in order to maintain the temporal accuracy. This is opposite to the common practice for the steady state convergence, and is believed to be due to the PBE nonlinearity and its time splitting treatment. Effective adaptive schemes have been constructed so that the pseudo-time GFM methods become more efficient than the constant $\Delta t$ ones.

A broken symmetry approach for the modeling and analysis of antiparallel diodes-based chaotic circuits: a case study

This paper focuses on the modeling and symmetry breaking analysis of the large class of chaotic electronic circuits utilizing an antiparallel diodes pair as nonlinear device necessary for chaotic oscillations. A new and relatively simple autonomous jerk circuit is used as a paradigm. Unlike current approaches assuming identical diodes (and thus a perfect symmetric circuit), we consider the more realistic situation where both antiparallel diodes present different electrical properties in spite of unavoidable scattering of parameters. Hence, the nonlinear component synthesized by the diodes pair exhibits an asymmetric current–voltage characteristic which engenders the explicit symmetry break of the whole electronic circuit. The mathematical model of the new circuit consists of a continuous time 3D autonomous system with exponential nonlinear terms. We investigate the chaos mechanism with respect to model parameters and initial conditions as well, both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectories plots as main indicators. We report period doubling route to chaos, spontaneous symmetry breaking, merging crisis and the coexistence of two, four, or six mutually symmetric attractors in the symmetric regime of operation. More intriguing and complex nonlinear behaviors are revealed in the asymmetric system such as coexisting asymmetric bubbles of bifurcation (a new kind of phenomenon discovered in this work), hysteretic dynamics, critical phenomena, and coexisting multiple (i.e. two, three, four, or five) asymmetric attractors for some appropriately selected values of parameters. Laboratory experimental tests are conducted to support the theoretical analysis. The results obtained in this work clearly indicate that chaotic circuits with back to back diodes can demonstrate much more complex dynamics than what is reported in the relevant literature and thus should be reconsidered accordingly following the method described in this paper.

Pseudo-transient ghost fluid methods for the Poisson-Boltzmann equation with a two-component regularization

The Poisson Boltzmann equation (PBE) is a well-established implicit solvent continuum model for the electrostatic analysis of solvated biomolecules. The numerical solution of the nonlinear PBE is still a challenge due to its exponential nonlinear term, strong singularity by the source terms, and distinct dielectric regions. In this paper, a new pseudo-transient approach is proposed, which combines an analytical treatment of singular charges in a two-component regularization, with an analytical integration of nonlinear term in pseudo-time solution. To ensure efficiency, both fully implicit alternating direction implicit (ADI) and unconditionally stable locally one-dimensional (LOD) methods have been constructed to decompose three-dimensional linear systems into one-dimensional (1D) ones in each pseudo-time step. Moreover, to accommodate the nonzero function and flux jumps across the dielectric interface, a modified ghost fluid method (GFM) has been introduced as a first order accurate sharp interface method in 1D style, which minimizes the information needed for the molecular surface. The 1D finite-difference matrix generated by the GFM is symmetric and diagonally dominant, so that the stability of ADI and LOD methods is boosted. The proposed pseudo-transient GFM schemes have been numerically validated by calculating solvation free energy, binding energy, and salt effect of various proteins. It has been found that with the augmentation of regularization and GFM interface treatment, the ADI method not only enhances the accuracy dramatically, but also improves the stability significantly. By using a large time increment, an efficient protein simulation can be realized in steady-state solutions. Therefore, the proposed GFM-ADI and GFM-LOD methods provide accurate, stable, and efficient tools for biomolecular simulations.

On chaotic models with hidden attractors in fractional calculus above power law

Abstract Researchers around the world are still wondering about the real origin and causes of hidden oscillating regimes and hidden attractors exhibited by some non-linear complex models. Such models are characterized by a dynamic with a basin of attraction that does not contain neighborhoods of equilibrium points. In this paper, we show that hidden oscillating regimes and hidden attractors can also exist in systems resulting from a combination with fractional differentiation. We apply a fractional derivative with Mittag–Leffler Kernel to a dynamical system with an exponential non-linear term and analyzed the resulting model both analytically and numerically. The combined model, which has no equilibrium points is however shown to display complex oscillating trajectories that culminate in chaos. Numerical simulations show some bifurcation dynamics with respect to the derivative order β and prove that the observed chaotic behavior persists as β varies. These observations made here allow us to say that the fractional model under study belongs to the category of systems with hidden oscillations.

Hopf bifurcation, antimonotonicity and amplitude controls in the chaotic Toda jerk oscillator: analysis, circuit realization and combination synchronization in its fractional-order form

ABSTRACTIn this paper, an autonomous Toda jerk oscillator is proposed and analysed. The autonomous Toda jerk oscillator is obtained by converting an autonomous two-dimensional Toda oscillator with an exponential nonlinear term to a jerk oscillator. The existence of Hopf bifurcation is established during the stability analysis of the unique equilibrium point. For a suitable choice of the parameters, the proposed autonomous Toda jerk oscillator can generate antimonotonicity, periodic oscillations, chaotic oscillations and bubbles. By introducing two additional parameters in the proposed autonomous Toda jerk oscillator, it is possible to control partially or totally the amplitude of its signals. In addition, electronic circuit realization of the proposed Toda jerk oscillator is carried out to confirm results found during numerical simulations. The commensurate fractional-order version of the proposed autonomous chaotic Toda jerk oscillator is studied using the stability theorem of fractional-order oscillators and numerical simulations. It is found that periodic oscillations and chaos exist in the fractional-order form of the proposed Toda jerk oscillator with order less than three. Finally, combination synchronization of two fractional-order proposed autonomous chaotic Toda jerk oscillators with another fractional-order proposed autonomous chaotic Toda jerk oscillator is analysed using the nonlinear feedback control method.

Dimensionality Reduction Reconstitution for Extreme Multistability in Memristor‐Based Colpitts System

In this paper, a four‐dimensional (4‐D) memristor‐based Colpitts system is reaped by employing an ideal memristor to substitute the exponential nonlinear term of original three‐dimensional (3‐D) Colpitts oscillator model, from which the initials‐dependent extreme multistability is exhibited by phase portraits and local basins of attraction. To explore dynamical mechanism, an equivalent 3‐D dimensionality reduction model is built using the state variable mapping (SVM) method, which allows the implicit initials of the 4‐D memristor‐based Colpitts system to be changed into the corresponding explicitly initials‐related system parameters of the 3‐D dimensionality reduction model. The initials‐related equilibria of the 3‐D dimensionality reduction model are derived and their initials‐related stabilities are discussed, upon which the dynamical mechanism is quantitatively explored. Furthermore, the initials‐dependent extreme multistability is depicted by two‐parameter plots and the coexistence of infinitely many attractors is demonstrated by phase portraits, which is confirmed by PSIM circuit simulations based on a physical circuit.

Dynamic analysis and synchronization control of an unusual chaotic system with exponential term and coexisting attractors

This paper reports a new four-dimensional chaotic system consisting of an exponential nonlinear term, two quadratic nonlinear terms and five linear terms. The system has only one equilibrium and performs stability, periodicity and chaos with the variation of the parameters. It losses its stability with the occurrence of Hopf bifurcation and goes into chaos via period-doubling bifurcation. One more interesting feature of the system is that it can generate multiple coexisting attractors for different initial conditions, such as two strange attractors with one limit cycle, one strange attractor with two limit cycles, etc. The dynamic properties of the system are presented by numerical simulation includes bifurcation diagrams, Lyapunov exponent spectrum and phase portraits. An electronic circuit is constructed to implement the chaotic attractor of the system. Based on the linear quadratic regulator (LQR) method, the synchronization control of the system is investigated.

Tracking control for a ten-ring chaotic system with an exponential nonlinear term

A new system with an exponential nonlinear term, which can exhibit ten-ring attractors, is introduced in this paper. The proposed chaotic system is totally different from the existing systems in previous works. Some basic dynamical properties of the system are analyzed theoretically and numerically. Existence and uniqueness of the solution of the presented system are proved. Next, the tracking control for the proposed system with exponential convergence rate is addressed. Finally, numerical simulations are also performed to verify the effectiveness of presented schemes.

A numerical study for optimal portfolio regime-switching model I. 2D Black–Scholes equation with an exponential non-linear term

A system of weakly coupled semilinear parabolic equations of optimal portfolio in a regime-switching model is suggested by Valdez and Vargiolu (2013). To study the effects from the typical features of the system, i.e. the boundary degeneration, exponential non-linearity and mixed derivative, we consider a representative 2D Black–Scholes semilinear equation model. We construct and analyze sign preserving, flux limited finite difference schemes for the scalar problem. Numerical experiments are discussed.

Hidden attractors in a chaotic system with an exponential nonlinear term

Studying systems with hidden attractors is new attractive research direction because of its practical and threoretical importance. A novel system with an exponential nonlinear term, which can exhibit hidden attractors, is proposed in this work. Although new system possesses no equilibrium points, it displays rich dynamical behaviors, like chaos. By calculating Lyapunov exponents and bifurcation diagram, the dynamical behaviors of such system are discovered. Moreover, two important features of a chaotic system, the possibility of synchronization and the feasibility of the theoretical model, are also presented by introducing an adaptive synchronization scheme and designing a digital hardware platform-based emulator.

On nonlinear Schrödinger equations derived from the nonrelativistic limit of nonlinear Klein–Gordon equations in de Sitter spacetime

Some Schrödinger equations with weighted nonlinear terms are derived from the nonrelativistic limit of nonlinear Klein–Gordon equations in de Sitter spacetime. Local and global solutions for the Cauchy problem are considered in Sobolev spaces for power type and exponential type nonlinear terms. The roles of spatial expansion and contraction on the problem are studied. And a dissipative property of the equations is remarked.

Amplitude control and projective synchronization of a dynamical system with exponential nonlinearity

A dynamical system with exponential nonlinearity is reported in this paper. Analysis shows that one can flexibly control signal amplitude of the system by introducing control function in the exponential nonlinear term, and the corresponding Lyapunov exponents keep invariable. But the coefficient in cross-product term cannot provide amplitude modulation. By considering the fact that the unique amplitude function can provide the scale factors, a control scheme combining the techniques of linear feedback and variable substitution is presented to realize projective synchronization of the chaotic system.

Remarks on global solutions of dissipative wave equations with exponential nonlinear terms

The Cauchy problem for dissipative wave equations with exponential type nonlinear terms is considered in the energy space in two spatial dimensions. The nonlinear terms have a singularity at the origin, and global solutions are shown based on the Gagliardo-Nirenberg type inequality.

Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms

The Cauchy problem of Klein-Gordon equations is considered for power and exponential type nonlinear terms with singular weights. Time local and global solutions are shown to exist in the energy class. The Caffarelli-Kohn-Nirenberg inequality and the Trudinger-Moser type inequality with singular weights are applied to the problem.

A Research on the Synchronization of Two Novel Chaotic Systems Based on a Nonlinear Active Control Algorithm

Abstract — The problem of chaos synchronization is to design a coupling between two chaotic systems (master-slave/drive-response systems configuration) such that the chaotic time evaluation becomes ideal and the output of the slave (response) system asymptotically follows the output of the master (drive) system. This paper has addressed the chaos synchronization problem of two chaotic systems using the Nonlinear Control Techniques, based on Lyapunov stability theory. It has been shown that the proposed schemes have outstanding transient performances and that analytically as well as graphically, synchronization is asymptotically globally stable. Suitable feedback controllers are designed to stabilize the closed-loop system at the origin. All simulation results are carried out to corroborate the effectiveness of the proposed methodologies by using Mathematica 9. Keywords-Synchronization; Lyapunov Stability Theory; Nonlinear Control; Routh-Hurwitz Criterion I. I NTRODUCTION Synchronization of chaotic systems is a process where two (or many) chaotic systems eventually progress identically for different initial conditions in all future states. This means that the dynamical state of one of the system is completely dictated by the dynamical state of the other system [1]. Chaos Synchronization between two chaotic systems is one of the most primary procedures in complex systems’ control and has wide potential applications in different fields [2-6]. After a pioneering work on chaos synchronization [1], synchronization of chaotic dynamical systems has received a great interest among researchers in nonlinear sciences for more than two decades [7]. Until now, diverse techniques have been proposed and applied successfully to synchronize two identical (or nearly identical) as well as nonidentical chaotic systems [8-13]. Notable among those, the Nonlinear control algorithm [7, 9] is one of the effectual techniques for synchronizing two chaotic systems [7]. Nonlinear control techniques take the advantage of the given nonlinear system dynamics to produce high-performance designs. No Lyapunov exponents or gain matrix are required for its execution. These qualities allow the designer to focus on the synchronization problem, leaving troublesome model manipulations [9]. Edward Lorenz, a meteorologist and mathematician, is known to be the pioneer of chaos theory. In the 1960s, Lorenz made his historical discovery by observing weather phenomena particularly in convections of fluids [14]. Lorenz took different mathematical models of fluid convection and simplified them into a system of ordinary differential equations and came up with a 3-D chaotic attractor for the first time, what is now known as the popular Lorenz equations [14]. After the exceptional discovery of E. Lorenz on chaotic attractor, chaos has become an interesting topic for many researchers. During the last three decades, remarkable research has been done on chaos which explored its different applications, features and fundamental properties [15]. The significance of the 3-D differential equations is that relatively simple systems could exhibit rather complex or specifically chaotic behavior. The 3-dimensional chaotic systems have many potential applications in different scientific fields such as chemical reactions, secure communications, biological systems and nonlinear circuits [15]. Due to a wide range of applications of 3-D chaotic systems, various systems such as the Chen system, Rossler system, Liu system, Qi system, Tigan system and Lu system [16-19] have been proposed and applied successfully to many practical systems and have shown some effective outcomes. Recently, a new 3-D autonomous chaotic system based on a quadratic exponential nonlinear term and a quadratic cross product term has been proposed and studied [20]. A quadratic exponential nonlinear term was added to the third equation while eliminating the second term from the second equation and a nonlinear term from the third equation of the Lorenz System [20]. The new 3-D chaotic system is topologically different from the Lorenz System. The two-scroll attractor from the new system exhibits multiplex chaotic dynamics. The nonlinear dynamical properties of the new

An energy-conserving second order numerical scheme for nonlinear hyperbolic equation with an exponential nonlinear term

We present a second order accurate numerical scheme for a nonlinear hyperbolic equation with an exponential nonlinear term. The solution to such an equation is proven to have a conservative nonlinear energy. Due to the special nature of the nonlinear term, the positivity is proven to be preserved under a periodic boundary condition for the solution. For the numerical scheme, a highly nonlinear fractional term is involved, for the theoretical justification of the energy stability. We propose a linear iteration algorithm to solve this nonlinear numerical scheme. A theoretical analysis shows a contraction mapping property of such a linear iteration under a trivial constraint for the time step. We also provide a detailed convergence analysis for the second order scheme, in the ℓ∞(0,T;ℓ∞) norm. Such an error estimate in the maximum norm can be obtained by performing a higher order consistency analysis using asymptotic expansions for the numerical solution. As a result, instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O(Δt3+h4) convergence in ℓ∞(0,T;ℓ2) norm, which leads to the necessary ℓ∞ error estimate using the inverse inequality, under a standard constraint Δt≤Ch. A numerical accuracy check is given and some numerical simulation results are also presented.

New solution decomposition and minimization schemes for Poisson–Boltzmann equation in calculation of biomolecular electrostatics

The Poisson–Boltzmann equation (PBE) is one widely-used implicit solvent continuum model in the calculation of electrostatic potential energy for biomolecules in ionic solvent, but its numerical solution remains a challenge due to its strong singularity and nonlinearity caused by its singular distribution source terms and exponential nonlinear terms. To effectively deal with such a challenge, in this paper, new solution decomposition and minimization schemes are proposed, together with a new PBE analysis on solution existence and uniqueness. Moreover, a PBE finite element program package is developed in Python based on the FEniCS program library and GAMer, a molecular surface and volumetric mesh generation program package. Numerical tests on proteins and a nonlinear Born ball model with an analytical solution validate the new solution decomposition and minimization schemes, and demonstrate the effectiveness and efficiency of the new PBE finite element program package.