Numerical technique of perturbation method for computing periodic solution of piecewise-linear autonomous system

1983 ◽  
Vol 66 (10) ◽  
pp. 1-10 ◽  
Author(s):  
Kohshi Okumura ◽  
Akira Kishima
Keyword(s):  
Periodic Solution ◽  
Perturbation Method ◽  
Autonomous System ◽  
Numerical Technique ◽  
Piecewise Linear ◽  
Linear Autonomous System

GENERATING 3-D SCROLL GRID ATTRACTORS OF FRACTIONAL DIFFERENTIAL SYSTEMS VIA STAIR FUNCTION

2007 ◽  
Vol 17 (11) ◽  
pp. 3965-3983 ◽  
Author(s):  
WEIHUA DENG
Keyword(s):  
Differential System ◽  
Autonomous System ◽  
Function Approach ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Linear Autonomous System ◽  
Fractional Differential ◽  
The One ◽  
Number Of Equilibria

This paper discusses the stair function approach for the generation of scroll grid attractors of fractional differential systems. The one-directional (1-D) n-grid scroll, two-directional (2-D) (n × m)-grid scroll and three-directional (3-D) (n × m × l)-grid scroll attractors are created from a fractional linear autonomous system with a simple stair function controller. Being similar to the scroll grid attractors of classical differential systems, the scrolls of 1-D n-grid scroll, 2-D (n × m)-grid scroll and 3-D (n × m × l)-grid scroll attractors are located around the equilibria of fractional differential system on a line, on a plane or in 3D, respectively and the number of scrolls is equal to the corresponding number of equilibria.

Uniformly Bounded Input Gives Ultimately Bounded Output: On the Dynamics of a Piecewise Linear Beam System

1999 ◽  
Author(s):  
Marcel F. Heertjes ◽  
Marinus J. G. van de Molengraft ◽  
Jan J. Kok
Keyword(s):  
Periodic Solution ◽  
Piecewise Linear ◽  
Wave Force ◽  
Degree Of Freedom ◽  
Harmonic Force ◽  
Periodic Force ◽  
Beam System ◽  
Bounded Input ◽  
Uniformly Bounded

Abstract A periodically excited piecewise linear beam system is studied. The beam system consists of a supported multi-degree-of-freedom beam with one-sided spring. This system is proved to have a 1-periodic solution to any uniformly bounded periodic force applied along the beam. The existence of a 1-periodic solution will be shown numerically and experimentally for both a harmonic force and a block-wave force.

GENERATING CHAOS VIA A DYNAMICAL CONTROLLER

2001 ◽  
Vol 11 (03) ◽  
pp. 865-869 ◽  
Author(s):  
GUO-QUN ZHONG ◽  
KIM F. MAN ◽  
GUANRONG CHEN
Keyword(s):  
Computer Simulation ◽  
Autonomous System ◽  
Quadratic Function ◽  
Feedback Controller ◽  
Nonlinear Feedback ◽  
Circuit Implementation ◽  
Linear Autonomous System

This Letter studies the generation of chaos from a linear autonomous system by employing a dynamical nonlinear feedback controller. The system setup is quite simple, and the only nonlinearity is a piecewise-quadratic function in the form of x|x|. Both computer simulation and circuit implementation are given to verify the chaos generated by this mechanism.

EFFECT OF FRACTIONAL DERIVATIVE PROPERTIES ON THE PERIODIC SOLUTION OF THE NONLINEAR OSCILLATIONS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050095 ◽  
Author(s):  
YUSRY O. EL-DIB ◽  
NASSER S. ELGAZERY
Keyword(s):  
Periodic Solution ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Nonlinear Oscillations ◽  
Duffing Oscillator ◽  
Solution Process ◽  
Auxiliary Parameter ◽  
Periodic Force ◽  
Definition Of ◽  
Perturbed Equation

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

Smoothness of Asymptotic Phase Revisited

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Flaviano Battelli ◽  
Kenneth J. Palmer
Keyword(s):  
Periodic Solution ◽  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stable Manifold ◽  
Autonomous System ◽  
Asymptotic Phase

AbstractIt is well-known that solutions on the stable manifold of a hyperbolic periodic solution of an autonomous system of ordinary differential equations have an asymptotic phase which has the same order of smoothness as the vector field. In this paper we show if the system depends on a parameter that, in general, the asymptotic phase loses one order of smoothness in the parameter.

scholarly journals CHAOS IN A NONAUTONOMOUS MODEL FOR THE INTERACTIONS OF PREY AND PREDATOR WITH EFFECT OF WATER LEVEL FLUCTUATION

2020 ◽  
Vol 28 (04) ◽  
pp. 865-900
Author(s):  
ABHIJIT SARKAR ◽  
PANKAJ KUMAR TIWARI ◽  
FRANCESCA BONA ◽  
SAMARES PAL
Keyword(s):  
Periodic Solution ◽  
Water Level ◽  
Autonomous System ◽  
Chaotic Behavior ◽  
Positive Periodic Solution ◽  
Predation Rate ◽  
Nonautonomous System ◽  
Almost Periodic ◽  
Artificial Lake ◽  
Stable Periodic

Water level regulates the dynamics of different populations residing in water bodies. The increase/decrease in the level of water leads to an increase/decrease in the volume of water, which influences the interactions of fishes and catching capability. We examine how seasonal variations in water level and harvesting affect the outcome of prey–predator interactions in an artificial lake. A seasonal variation of the water level is introduced in the predation rate. We derive conditions for the persistence and extinction of the populations. Using the continuation theorem, we determine the conditions for which the system has a positive periodic solution. The existence of a unique globally stable periodic solution is also presented. Moreover, we obtain conditions for the existence, uniqueness and stability of a positive almost periodic solution. We find that if the autonomous system has a stable focus, the corresponding nonautonomous system exhibits a unique stable positive periodic solution. But, whenever the autonomous system shows limit cycle oscillations, the corresponding nonautonomous system exhibits chaotic dynamics. The chaotic behavior of system is confirmed by the positivity of the maximal Lyapunov exponent. For higher values of the assimilation fraction of prey population, the persistent oscillations of the autonomous system are eliminated and this system becomes stable. On the other hand, chaotic nature of the nonautonomous system is converted into periodicity if the assimilation fraction of prey is large. Moreover, populations behave almost periodically if the seasonally varied rate parameters are almost periodic functions of time. Our findings show that water level plays an important role in the persistence of prey–predator system.

scholarly journals A Chaotic Three Dimensional Non-Linear Autonomous System Beyond Lorenz Type Systems

2012 ◽  
Vol 36 (2) ◽  
pp. 159-170
Author(s):  
Md Shariful Islam Khan ◽  
Md Shahidul Islam
Keyword(s):  
Autonomous System ◽  
Three Dimensional ◽  
Type Systems ◽  
Control Parameters ◽  
Fundamental Properties ◽  
Non Linear ◽  
Linear Autonomous System ◽  
Non Linear System ◽  
Academy Of Sciences ◽  
The Relationship

Some fundamental properties of a chaotic three-dimensional non-linear system of the Lorenz type systems were studied. The invariance, dissipation, bifurcation and the strange attractors were investigated and analyzed one 1-scroll, two 2-scroll and two 4-scroll attractors by adding control   parameters to this system. The relationship and connecting function for the 2-scroll attractor of this system were also explored. DOI: http://dx.doi.org/10.3329/jbas.v36i2.12959 Journal of Bangladesh Academy of Sciences, Vol. 36, No. 2, 159-170, 2012  

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