Orbits Of Nonlinear Dynamical Systems Research Articles

Bifurcations of phase portraits and chaotic behaviors of the (2+1)-dimensional double‐chain DNA system with beta derivative: A qualitative approach

Qualitative analysis in mathematical modeling has become an important research area within the broad domain of nonlinear sciences. In the realm of qualitative analysis, the bifurcation method is one of the significant approaches for studying the structure of orbits in nonlinear dynamical systems. To apply the bifurcation method to the (2 + 1)-dimensional double-chain Deoxyribonucleic Acid system with beta derivative, the bifurcations of phase portraits and chaotic behaviors, combined with sensitivity and multi-stability analysis of this system, are examined. Initially, the bifurcations of phase portraits are visually identified at the obtained equilibrium points of a planar dynamical system via both Hamiltonian and Jacobian algorithms. The obtained results indicate Jacobian algorithm is more efficient in identifying the stability of bifurcations than the Hamiltonian algorithm for this system. Subsequently, by introducing an external perturbation term into the planar dynamical system, the chaotic behavior is effectively identified by using a variety of tools, such as two- and three-dimensional phase portraits, time series, Lyapunov exponents, and Poincaré maps. The findings suggest that the perturbed dynamical system deviates from regular patterns and exhibits behavior ranging from periodic to quasi-periodic and from quasi-periodic to chaotic. Finally, the sensitivity and multi-stability of the system are examined using the Runge-Kutta method to assess the model's response to minor variations in initial conditions through numerical solutions, revealing that the model is sensitive and multi-stable. The outcomes of this study will enhance a relationship between applied mathematicians and experimental biologists, helping to explore hidden features of Deoxyribonucleic Acid through the studied model.

Jacobian-free variational method for computing connecting orbits in nonlinear dynamical systems.

One approach for describing spatiotemporal chaos is to study the unstable invariant sets embedded in the chaotic attractor of the system. While equilibria, periodic orbits, and invariant tori can be computed using existing methods, the numerical identification of heteroclinic and homoclinic connections between them remains challenging. We propose a robust matrix-free variational method for computing connecting orbits between equilibrium solutions. Instead of a common shooting-based approach, we view the identification of a connecting orbit as a minimization problem in the space of smooth curves in the state space that connect the two equilibria. In this approach, the deviation of a connecting curve from an integral curve of the vector field is penalized by a non-negative cost function. Minimization of the cost function deforms a trial curve until, at a global minimum, a connecting orbit is obtained. The method has no limitation on the dimension of the unstable manifold at the origin equilibrium and does not suffer from exponential error amplification associated with time-marching a chaotic system. Owing to adjoint-based minimization techniques, no Jacobian matrices need to be constructed. Therefore, the memory requirement scales linearly with the size of the problem, allowing the method to be applied to high-dimensional dynamical systems. The robustness of the method is demonstrated for the one-dimensional Kuramoto-Sivashinsky equation.

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.

4 Boyutlu Modelde Spinör Tipi İnstanton Çözümlerinin Dalgacık Entropisinin İncelenmesi

In recent years, many methods have been proposed in various fields to investigate the properties of orbits of nonlinear dynamical systems. In this study, the characteristics of the orbits of spinor-type instanton solutions in the four-dimensional Gursey model via Heisenberg ansatz is investigated. The orbits of spinor-type instanton solutions are analyzed by Shannon wavelet entropy (WE) method. In order to analyze the regular or irregular states of the orbits of spinor-type instanton solutions, WE spectrum and WE in phase space are studied. It is observed that spinor-type instanton solutions have regular orbits around the fixed point and irregular orbits for other points. According to this study, WE can be used to observe the entropy evolution of nonlinear dynamical systems in phase space.

State-dependent act-and-wait time-delayed feedback control algorithm

Time-delayed feedback control (TDFC) method is one of the most popular experimental tools used for stabilization of unstable periodic orbits in nonlinear dynamical systems. The theory of the method is difficult since systems controlled by time-delayed feedback evolve in an infinite-dimensional function space. Here we modify the TDFC by appending the feedback loop with a state-dependent switch. The switch closes and opens the feedback loop in the dependence of the phase of the output signal, which is measured by a specially designed phase detector. This modification represents a further development of the act-and-wait concept applied to time-delayed feedback controllers. The concept allows to reduce the phase space dimension of time-delayed feedback systems and to significantly simplify the theory. In previous works, the time-delayed feedback loop was tuned on and off periodically, independently of the state of the system. We discuss the advantages of our algorithm with respect to previously considered modifications. The efficacy of the proposed algorithm is numerically demonstrated with autonomous Rössler and Lorenz systems as well as a nonautonomous Duffing oscillator.

Approximate Prediction-Based Control Method for Nonlinear Oscillatory Systems with Applications to Chaotic Systems

The approximate Prediction-Based Control method (aPBC) is the continuous-time version of the well-known Prediction-Based Chaos Control method applied to stabilize periodic orbits of nonlinear dynamical systems. The method is based on estimating future states of the free system response of continuous-time systems using the solution from the Runge-Kutta implicit method in real time. Some aspects of aPBC are evaluated in the present work, particularly its robustness to low future states estimation precision is exemplified.

Optimized shooting method for finding periodic orbits of nonlinear dynamical systems

An alternative numerical method is developed to find stable and unstable periodic orbits of nonlinear dynamical systems. The method exploits the high efficiency of the Levenberg–Marquardt algorithm for medium-sized problems and has the additional advantage of being relatively simple to implement. It is also applicable to both autonomous and non-autonomous systems. As an example of its use, it is employed to find periodic orbits in the Rössler system, a coupled Rössler system, as well as an eight-dimensional model of a flexible rotor-bearing; problems which have been treated previously via two related methods. The results agree with the previous methods and are seen to be more accurate in some cases. A simple implementation of the method, written in the Python programming language, is provided as an Appendix.

A variational approach to connecting orbits in nonlinear dynamical systems

We propose a variational method for determining homoclinic and heteroclinic orbits including spiral-shaped ones in nonlinear dynamical systems. Starting from a suitable initial curve, a homotopy evolution equation is used to approach a true connecting orbit. The procedure is an extension of a variational method that has been used previously for locating cycles, and avoids the need for linearization in search of simple connecting orbits. Examples of homoclinic and heteroclinic orbits for typical dynamical systems are presented. In particular, several heteroclinic orbits of the steady-state Kuramoto–Sivashinsky equation are found, which display interesting topological structures, closely related to those of the corresponding periodic orbits. • A variational technique for determining connecting orbits is proposed. • It does not need linearization for searching simple connecting orbits. • The initialization and parameter evolution becomes particularly simple. • New connections of the steady-state Kuramoto–Sivashinsky equation are found.

Floquet stability analysis of Ott–Grebogi–Yorke and difference control

Stabilization of instable periodic orbits of nonlinear dynamical systems has been a widely explored field theoretically and in applications. The techniques can be grouped into time-continuous control schemes based on Pyragas, and the two Poincaré-based chaos control schemes, Ott–Gebogi–Yorke (OGY) and difference control. Here, a new stability analysis of these two Poincaré-based chaos control schemes is given by means of the Floquet theory. This approach allows to calculate exactly the stability restrictions occurring for small measurement delays and for an impulse length shorter than the length of the orbit. This is of practical experimental relevance; to avoid a selection of the relative impulse length by trial and error, it is advised to investigate whether the used control scheme itself shows systematic limitations on the choice of the impulse length. To investigate this point, a Floquet analysis is performed. For OGY control the influence of the impulse length is marginal. As an unexpected result, difference control fails when the impulse length is taken longer than a maximal value that is approximately one half of the orbit length for small Ljapunov numbers and decreases with the Ljapunov number.

ANALYTICAL CONSTRUCTION OF HOMOCLINIC ORBITS OF TWO- AND THREE-DIMENSIONAL DYNAMICAL SYSTEMS

An approach for the construction of homoclinic orbits of non-linear dynamical systems with phase spaces of dimensions equal to two or three is proposed here. The non-linear Schrodinger equation and Lorenz system are considered. Quasi-Pade' approximants are used for this construction. Potentiality and convergence conditions used earlier in the theory of non-linear normal vibration modes make it possible to solve the boundary-value problems formulated for the orbits and to evaluate initial amplitude values.

On periodic orbits of nonlinear dynamical systems with many degrees of freedom

The purpose of this article is to extend our investigation to periodic orbits of nonlinear dynamical systems with many degrees of freedom. The method of generalized averaging and the indicatrix method are developed. An example is studied to illustrate our techniques. The analytical expressions are in good agreement with numerical results computed by the indicatrix method for this example.

A new method for constructing periodic orbits in nonlinear dynamicalsystems.

An iterative method for constructing periodic orbits in nonlinear dynamical systems is developed. The method is a modification of the Generalized Newton-Raphson technique. Application of the method to numerically continuing natural and isoperiodic families in the restricted three-body problem was implemented on a high-speed computer and the results show broad regions of convergence, lack of sensitivity, and strong convergence properties of the method. Previous methods isolated periodic orbits by adjusting the initial conditions and the period of a solution of the dynamical system until the solution closed to a prescribed accuracy. The new technique is distinctly different in that it constructs a sequence of precisely periodic solutions of a system of linear differential equations which approximates the original dynamical system. This system of linear differential equations is modified at each step to more closely represent the dynamical system. The technique gives a way for both design of starting orbits for detailed study of orbit families with the methods of initial condition adjustment and for construction of elements of new orbit families. It is applicable to nonautonomous systems and a convergence theorem for this case is presented. Although the technique was applied with success to autonomous systems, a convergence theorem for the time-free case was not established. It is shown that the method will approach the desired solution in this case arbitrarily closely for computational purposes.