Chaos in fractional-order autonomous nonlinear systems
Summary (2 min read)
1. Introduction
- Chaotic systems have been a focal point of renewed interest for many researchers in the past few decades.
- In their study the authors will focus on two types of autonomous chaotic systems that are of practical interest: an electronic chaotic oscillator model [10], and a mechanical chaotic ‘‘jerk’’ model [11,12].
- The authors derive the approximate linear transfer functions for the fractional integrator of order that varies from 0.1 to 0.9, and study the resulting behavior of the entire system for each case under the effect of different types of nonlinearities.
- The authors will follow the algorithm in [13] to calculate linear transfer function approximations of (1).
2.1. Fractional chaotic oscillator
- The authors consider the one-parameter, third-order chaotic oscillator of canonical structure reported in [10] as being the simplest possible structure for a chaotic oscillator.
- The output in- tegrator of this model is then replaced with a fractional integrator.
3. Transfer function approximations
- Using the algorithm in [11], Table 1 gives the resulting approximating transfer functions, HðsÞ for different fractional orders, in increments of 0.1, for the mth order output integrator, assuming xmax ¼ 100 and pT ¼ 0:01.
- Notice that the order of the approximate transfer functions increases as the desired error decreases.
- The chaotic systems can then be represented by block diagrams as shown in Fig. 1a and b. In Fig. 1c, the authors show the magnitude Bode diagrams for a fractional integrator of order m ¼ 0:5, and its linear approximating transfer function (from Table 1).
- It can be seen that within the bandwidth of interest the two diagrams are in good agreement.
4. Simulation results
- In their simulations, the authors have visually inspected the bifurcation diagrams to identify chaos.
- These figures demonstrate clearly that indeed the chaotic behavior of the oscillator is not destroyed by order reduction.
- Also, it can be noticed that the effective chaotic range of the control parameter shifts downward away from that corresponding to the integer order model.
- In Figs. 9–11, the authors show simulation results of the fractional jerk model of total order of 2.9 and using the nonlinearities 9a, 9b, and 9c.
- From the above results, the authors present the following conjecture: Conjecture.
5. Conclusion
- The authors have demonstrated via numerical simulations that chaotic autonomous nonlinear systems can still exhibit chaotic behaviors when the order becomes fractional.
- Linear approximation techniques which are based on frequency domain arguments have been used to obtain approximate linear models of the given systems.
- The resulting order of the approximating function of the fractional integrator depends on the desired bandwidth, and error between actual and approximate Bode magnitude plots of the fractional integrator.
- Both electronic chaotic oscillators and chaotic jerk models were studied.
- The effective chaotic range of the control parameter for either model shrinks and moves downward as the total system order decreases and approaches 2, i.e. the chaotic range is affected by contraction and translation processes as the fractional order decreases.
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"Chaos in fractional-order autonomou..." refers methods in this paper
...According to the Poincare–Bendixon theorem [4], an integer order chaotic nonlinear system must have a minimum order of 3 for chaos to appear....
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"Chaos in fractional-order autonomou..." refers background or methods in this paper
...By utilizing frequency domain techniques based on Bode diagrams, one can obtain a linear approximation for the fractional-order integrator, the order of which depends on the desired bandwidth and discrepancy between the actual and the approximate magnitude Bode diagrams [13]....
[...]
...We will follow the algorithm in [13] to calculate linear transfer function approximations of (1)....
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...According to [13], if the discrepancy between the actual and approximate lines is specified as y dB over a frequency range of xmax, and for a corner frequency pT , then Eq....
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Frequently Asked Questions (18)
Q2. What is the effect of the control parameter on the fractional order?
The effective chaotic range of the control parameter for either model shrinks and moves downward as the total system order decreases and approaches 2, i.e. the chaotic range is affected by contraction and translation processes as the fractional order decreases.
Q3. How can one obtain a linear approximation for the fractional-order integrator?
By utilizing frequency domain techniques based on Bode diagrams, one can obtain a linear approximation for the fractional-order integrator, the order of which depends on the desired bandwidth and discrepancy between the actual and the approximate magnitude Bode diagrams [13].
Q4. What is the resulting order of the approximating function of the fractional integrator?
The resulting order of the approximating function of the fractional integrator depends on the desired bandwidth, and error between actual and approximate Bode magnitude plots of the fractional integrator.
Q5. What is the recent study of a wilson bridgeoscillator?
Wien bridgeoscillators have recently been studied in [9], where it is shown that limit cycles can be attained for any fractional order, with the proper value of the amplifier gain which, of course, is a function of the fractional order.
Q6. What is the effect of the fractional order on the chaotic range?
as the fractional order decreases below m ¼ 0:5, the chaotic range shrinks significantly, and becomes much less pronounced than that for the chaotic oscillator as shown in the figure.
Q7. What is the simplest way to show the chaotic system?
The integer order canonical chaotic oscillator with nonlinearity given by f ðx1Þ ¼ sgnðx1Þ is known to give a doublescroll-like chaotic attractor for control parameter values in the range 1:0 > a > 0:49.
Q8. What is the effect of the control parameter on the system?
it can be noticed that the effective chaotic range of the control parameter shifts downward away from that corresponding to the integer order model.
Q9. What is the effect of the control parameter on the chaotic range of the fractional integrator?
A third-order chaotic autonomous nonlinear system, with the appropriate nonlinearity and control parameters, is chaotic for any fractional order 2þ e, 1 > e > 0.
Q10. What is the effect of the chaotic behavior of the fractional-order models?
The chaotic behaviors of the fractional-order models exhibit some degree of qualitative robustness against model order reduction as would be caused by modeling errors.
Q11. How many numerical simulations have been conducted on fractional systems?
the authors have conducted several numerical simulations on fractional systems with total system order less than 2, as would be obtained by replacing all three integrators by fractional ones.
Q12. How many orders of chaos must exist for a nonlinear system to appear?
According to the Poincare–Bendixon theorem [4], an integer order chaotic nonlinear system must have a minimum order of 3 for chaos to appear.
Q13. What is the underlying theory behind fractional-order systems?
From a state-space point of view, the fractional order of the resulting characteristic equation results from a fractional-order integrator placed, for example, in place of the output integrator in the system block diagram realization.
Q14. What is the simplest possible structure for a chaotic oscillator?
The authors consider the one-parameter, third-order chaotic oscillator of canonical structure reported in [10] as being the simplest possible structure for a chaotic oscillator.
Q15. What is the control parameter for the fractional chaotic oscillator?
The resulting fractional chaotic oscillator model is then given in state space asdxm1 dtm¼ x2 dx2 dt¼ x3 dx3 dt ¼ aðx1 þ x2 þ x3 þ f ðx1ÞÞð7Þwhere a is the control parameter for this oscillator, and f ðx1Þ ¼ sgnðx1Þ is the model nonlinearity.
Q16. What is the simplest model for a chaotic oscillator?
This model, shown in Fig. 1b, is used to determine the time derivative of acceleration of an object, referred to as jerk[3,11,12].
Q17. How does the fractional jerk model behave?
In Figs. 12 and 13, the authors show simulations of the fractional jerk model of total orders 2.5 and 2.1, using nonlinearity 9c, and control values of 0.3 and 0.05, respectively.
Q18. What is the simplest way to study the behavior of a fractional integrator?
The authors derive the approximate linear transfer functions for the fractional integrator of order that varies from 0.1 to 0.9, and study the resulting behavior of the entire system for each case under the effect of different types of nonlinearities.