Stabilizing Periodic Orbits of the Fractional Order Chaotic Van Der Pol System

Unstable Periodic Orbit ◽

Van Der Pol ◽

Linear Feedback Controller

In this paper, stabilizing the unstable periodic orbits (UPO) in a chaotic fractional order system called Van der Pol is studied. Firstly, a technique for finding unstable periodic orbit in chaotic fractional order systems is stated. Then by applying this technique to the van der Pol system, unstable periodic orbit of system is found. After that, a method is presented for stabilization of the discovered UPO based on theories stability of the linear integer order and fractional order systems. Finally, a linear feedback controller was applied to the system and simulation is done for demonstration of controller performance.

Chaotic Control in a Fractional-Order Modified Van Der Pol Oscillator

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.474-476.83 ◽

2011 ◽

Vol 474-476 ◽

pp. 83-88

Author(s):

Xin Gao

Keyword(s):

Fractional Order ◽

Feedback Controller ◽

Van Der Pol Oscillator ◽

Linear Feedback ◽

Van Der Pol ◽

Chaotic Control ◽

Linear Feedback Controller

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we study the chaotic behaviors in a fractional-order modified van der Pol oscillator. We find that chaos exists in the fractional-order modified van der Pol oscillator with order less than 3. In addition, the lowest order we find for chaos to exist in such system is 2.4. Finally, a simple, but effective, linear feedback controller is also designed to stabilize the fractional order chaotic van der Pol oscillator.

Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems

Nonlinear Processes in Geophysics ◽

10.5194/npg-15-675-2008 ◽

2008 ◽

Vol 15 (4) ◽

pp. 675-680 ◽

Cited By ~ 2

Author(s):

Y. Saiki ◽

M. Yamada

Keyword(s):

Fluid Dynamics ◽

Periodic Orbit ◽

Periodic Orbits ◽

Space Physics ◽

Unstable Periodic Orbit ◽

Geophysical Fluid Dynamics ◽

Chaotic Orbits ◽

Low Dimensional ◽

Complex Phenomena

Abstract. Unstable periodic orbit (UPO) recently has become a keyword in analyzing complex phenomena in geophysical fluid dynamics and space physics. In this paper, sets of UPOs in low dimensional maps are theoretically or systematically found, and time averaged properties along UPOs are studied, in relation to those of chaotic orbits.

Improved Estimations of Unstable Periodic Orbits Extracted From Chaotic Sets

Volume 6C: 18th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc2001/vib-21585 ◽

2001 ◽

Author(s):

Z. Al-Zamel ◽

B. F. Feeny

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Duffing Oscillator ◽

Unstable Periodic Orbit ◽

Affine Map ◽

Saddle Type ◽

Recurrence Method

Abstract Unstable periodic orbits of the saddle type are often extracted from chaotic sets. We use the recurrence method of extracting segments of the chaotic data to approximate the true unstable periodic orbit. Then nearby trajectories are then examined to obtain the dynamics local to the extracted orbit, in terms of an affine map. The affine map is then used to estimate the true orbit. Accuracy is evaluated in examples including well known maps and the Duffing oscillator.

Dynamic Stabilization of Unstable Periodic Orbits in a CO2 Laser by Slow Modulation of Cavity Detuning

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127498001492 ◽

1998 ◽

Vol 08 (09) ◽

pp. 1783-1789 ◽

Cited By ~ 6

Author(s):

A. N. Pisarchik ◽

R. Corbalán ◽

V. N. Chizhevsky ◽

R. Vilaseca ◽

B. F. Kuntsevich

Keyword(s):

Periodic Orbit ◽

Periodic Orbits ◽

Transient Response ◽

Co2 Laser ◽

Dynamic Stabilization ◽

Dynamical Regime ◽

Unstable Periodic Orbit ◽

Periodic Attractor ◽

Slow Modulation

We demonstrate numerically and experimentally that a slow modulation of cavity detuning in a loss-modulated CO 2 laser can stabilize unstable periodic orbits even when the system remains in a particular dynamical regime for adiabatic changes of the detuning. When the parameter changes faster than the transient response of deformation of the original periodic attractor, the system can evolve toward an unstable periodic orbit.

CHAINS OF ROTATIONAL TORI AND FILAMENTARY STRUCTURES CLOSE TO HIGH MULTIPLICITY PERIODIC ORBITS IN A 3D GALACTIC POTENTIAL

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127411029823 ◽

2011 ◽

Vol 21 (08) ◽

pp. 2331-2342 ◽

Cited By ~ 9

Author(s):

M. KATSANIKAS ◽

P. A. PATSIS ◽

A. D. PINOTSIS

Keyword(s):

Phase Space ◽

Hamiltonian System ◽

Periodic Orbit ◽

Periodic Orbits ◽

Color Variation ◽

Space Structures ◽

Unstable Periodic Orbit ◽

Surface Of Section ◽

Autonomous Hamiltonian System

This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis & Zachilas, 1994] in order to visualize them. As examples, we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition, we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.

Optimization approach to the constrained regulation problem for linear continuous-time fractional-order systems

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0267 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Xindong Si ◽

Hongli Yang

Keyword(s):

Feedback Control ◽

Fractional Order ◽

State Feedback Control ◽

Invariant Sets ◽

Linear Feedback ◽

Control Law ◽

Optimization Approach ◽

Linear Feedback Control ◽

Computational Point

AbstractThis paper deals with the Constrained Regulation Problem (CRP) for linear continuous-times fractional-order systems. The aim is to find the existence conditions of linear feedback control law for CRP of fractional-order systems and to provide numerical solving method by means of positively invariant sets. Under two different types of the initial state constraints, the algebraic condition guaranteeing the existence of linear feedback control law for CRP is obtained. Necessary and sufficient conditions for the polyhedral set to be a positive invariant set of linear fractional-order systems are presented, an optimization model and corresponding algorithm for solving linear state feedback control law are proposed based on the positive invariance of polyhedral sets. The proposed model and algorithm transform the fractional-order CRP problem into a linear programming problem which can readily solved from the computational point of view. Numerical examples illustrate the proposed results and show the effectiveness of our approach.

Fractional-order Kalman filters for continuous-time fractional-order systems involving correlated and uncorrelated process and measurement noises

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331218790786 ◽

2018 ◽

Vol 41 (7) ◽

pp. 1933-1947 ◽

Cited By ~ 2

Author(s):

Fanghui Liu ◽

Zhe Gao ◽

Chao Yang ◽

Ruicheng Ma

Keyword(s):

Fractional Order ◽

Continuous Time ◽

Kalman Filters ◽

Equation Model ◽

Order System ◽

Derivative Method ◽

Measurement Noises ◽

Computation Load ◽

Average Derivative

This paper presents fractional-order Kalman filters using the fractional-order average derivative method for linear fractional-order systems involving process and measurement noises. By using the fractional-order average derivative method, a difference equation model is obtained by discretizing the investigated continuous-time fractional-order system, and the accuracy of state estimation is improved. Meanwhile, compared with the Tustin generating function, the fractional-order average derivative method proposed in this paper can reduce computation load and save calculation time. Two kinds of fractional-order Kalman filters are given, for the correlated and uncorrelated cases, in terms of the process and measurement noises for linear fractional-order systems, respectively. Finally, simulation results are illustrated to verify the effectiveness of the proposed Kalman filters using the fractional-order average derivative method.

Volume 9: 15th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications ◽

Model Predictive Control of General Fractional Order Systems Using a General Purpose Optimal Control Problem Solver: RIOTS

10.1115/detc2019-97489 ◽

2019 ◽

Author(s):

Sina Dehghan ◽

Tiebiao Zhao ◽

YangQuan Chen ◽

Taymaz Homayouni

Keyword(s):

Optimal Control ◽

Model Predictive Control ◽

Fractional Order ◽

Predictive Control ◽

Order System ◽

Problem Solver ◽

Application Process ◽

Order Model ◽

Integer Order

Abstract RIOTS is a Matlab toolbox capable of solving a very general form of integer order optimal control problems. In this paper, we present an approach for implementing Model Predictive Control (MPC) to control a general form of fractional order systems using RIOTS toolbox. This approach is based on time-response-invariant approximation of fractional order system with an integer order model to be used as the internal model in MPC. The implementation of this approach is demonstrated to control a coupled MIMO commensurate fractional order model. Moreover, the performance and its application process is compared to examples reported in the literature.