1. Chaotic Dynamics of the Muthuswamy-Chua System1
Workshop on Nonlinear Electronics And Its Applications
Vellore Institute of Technology, Vellore, India
August 31st 2013
Bharathwaj “Bart” Muthuswamy
Assistant Professor of Electrical Engineering
Milwaukee School of Engineering
BS (2002), MS (2005) and PhD (2009) from the University of California, Berkeley
PhD Advisor: Dr. Leon O. Chua (co-advised by Dr. Pravin P. Varaiya)
1“On the Integrability of a Muthuswamy-Chua System”, Llibre, J. and Valls, C. Journal of Nonlinear Mathematical Physics, Vol. 19, No. 4, pp – December 2012.

2. Nonlinear Dynamical Systems and Embedded Systems
What do I work on?
Nonlinear Dynamical Systems and Embedded Systems
- Applications and Mathematical properties of the Muthuswamy-Chua system
- Potential Applications to Turbulence Modeling
(MSOE; University of Western Australia, Perth, Australia)
- Chaotic Hierarchy and Flow Manifolds
(MSOE; University of Western Australia, Perth, Australia; I.U.T. de Toulon, La Garde Cedex, France)
- Applications of Chaotic Delay Differential Equations using Field Programmable Gate Arrays (FPGAs)
(MSOE; Vellore Institute of Technology; University Putra Malaysia, Malaysia; Springer-Verlag )
- Pattern Recognition Using Cellular Neural Networks on FPGAs
(MSOE; Altera Corporiation)
- Practical Memristors: discharge tubes, PN junctions and Josephson Junctions
(MSOE; IIT Chennai; University of Western Australia, Perth, Australia;
Vellore Institute of Technology, Vellore, India)
Education
- Nonlinear Dynamics at the undergraduate level (with folks from all over the world  )

3. Goal of This Talk Obtain chaos in the circuit [6] below:
So, the goal of this talk is to derive a circuit with the topology shown below. Also, only one of the circuit elements is active. Turns out that circuit is simply an “LCM” circuit: inductor, capacitor, memristor in series.
BUT a memristor is commercially unavailable, hence the emulator uses a LOT of analog components. The PURPOSE of this talk is primarily to go through interesting properties of this Muthuswamy-Chua circuit.
NOTE: WHEN YOU HAVE QUESTIONS, PLEASE STOP ME. THIS TALK IS MORE OF A DISCUSSION.

4. Outline I. Prerequisites for understanding this talk:
1. First course in circuit theory
2. First course in differential equations
II. Background
1. Introduction to Chaos
2. The fundamental circuit elements
III. The Memristor
1. Properties of the memristor
2. Memristive Devices
IV. The Muthuswamy-Chua circuit (System)
1. Derivation of Circuit Equations
2. Hardware emulation of the memristor
3. MultiSim (SPICE) simulation
4. An attractor from the circuit
5. Rigorous Mathematical Analysis
6. A Chaotic Hierarchy
7. Potential conservative chaotic systems
V. Conclusions, Current (future) work and References
So, lets look at an outline of this talk. First, the prerequisites to understanding this talk are:
1: Understand DC circuit theory, 2. Understand first order ODEs with constant coefficients
Then, we will look at a brief history of chaos and background:
II 1. We will mention Lorenz’s equation [7]. Also, Poincare-Bendixson theorem [1], [4], [9].
2. We will go over the four fundamental circuit elements [2].
1. Note that HP announced a memristor in 2008, we won’t go into the details of this device. For more information, please refer to [10].
2. Talk about general memristive devices [3].
IV. Talk about the Muthuswamy-Chua system [6]: discuss derivation of system equations and nonlinearities, show several attractors (will NOT illustrate local activity via the memristor's DC v-i characteristic).

5. Introduction to Chaos “Birth” of Chaos: Lorenz Attractor [7]
Edward Lorenz introduced the following nonlinear system of differential equations as a crude model of weather in 1963:
(1)
I put “Birth” in quotes because chaos was observed before Lorenz by a number of scientists (prominent: Poincare (led to Poincare map) and Van Der Pol (in his oscillator, dismissed it as “noise”)). But Lorenz was the first person to publish a paper on it (a very readable paper). He also identifies chaos in a simplified model of weather (practical system). But system was difficult to realize practically – can chaos exist in a physical system? Answer is yes. In order to build circuits, we need to understand circuit elements. But first, something important about the Lorenz system: it is autonomous AND three dimensions. Turns out that as a consequence of the Poincare-Bendixson theorem, three is the MINIMUM number of dimensions required for a continuous time system to become chaotic. Remember this!!!!!!!
Lorenz discovered that model dynamics were extremely sensitive to initial conditions and the trajectories were aperiodic but bounded.
But, does chaos exist physically? Answer is: YES. For example, chaotic circuits by Sprott [8].

6. The Fundamental Circuit Elementsv q i
We are all familiar with the fundamental circuit variables of voltage and current. The element that relates these two variables is the resistor using a relationship between dv and di (incremental resistance to allow for nonlinear resistors). Similarly, there is the capacitor and inductor. But, based on this picture there should be an element that relates electric flux to electric charge: memristor. This element was postulated by Leon in 1971 for “completeness” sake. Φ
Memristors were first postulated by Leon. O Chua in 1971 [2]

7. Outline I. Prerequisites for understanding this talk:
1. First course in circuit theory
2. First course in differential equations
II. Background
1. Introduction to Chaos
2. The fundamental circuit elements
III. The Memristor
1. Properties of the memristor
2. Memristive Devices
IV. The Muthuswamy-Chua circuit (System)
1. Derivation of Circuit Equations
2. Hardware emulation of the memristor
3. MultiSim (SPICE) simulation
4. An attractor from the circuit
5. Rigorous Mathematical Analysis
6. A Chaotic Hierarchy
7. Potential conservative chaotic systems
V. Conclusions, Current (future) work and References

8. Properties of the Memristor [2]
Circuit symbol:
A memristor defines a relation of the from:
If g is a single-valued function of charge (flux), then the memristor is
charge-controlled (flux-controlled)
Memristor i-v relationship:
M(q(t)) is the incremental memristance
Q1: Why is the memristor called “memory resistor”?
Because of the definition of memristance:
So, lets look at some properties of the memristor from Chua’s 1971 paper. The circuit symbol of the memristor is shown. So, a memristor defines a relation of the form: g(phi,q) = 0. If g is a single-valued function of charge, then the memristor is charge-controlled and vice-versa. Now, we can obtain an i-v relationship for a memristor as follows: M(q(t)) is defined as the incremental memristance or the slope of the phi-q curve. Two questions: Q1: Why is the memristor called “memory resistor”. Bart says: Because v(t)/i(t) = M(q(t)) = M(int_{-infty}^{t}i(tau)dtau). So, the memristance depends on past values of current flowing through the device: memory. Q2: Why is (KEYWORD) the memristor not relevant in linear circuit theory? Bart says: Two reasons. One, if the memristance is a constant (i.e..the slope of the phi-q is a straight-line), the v(t) = R.i(t), the memristor is simply a resistor. Second, the memristor (if memristance is not constant) is inherently a nonlinear device (you are sure to have heard of this term) in the sense of superposition: v(t) = M(int(i1+i2)).(i1+i2)(t) = M(int(i1)+int(i2)).(i1+i2)(t) need NOT equal M(int(i1).i1(t) + M(int(i2))i2(t).
But memristors were ignored for 40 years, until 2008 when HP announced the discovery of a physical memristor.
The memristor was defined in 1971, it was only in 2008 that HP was able to synthesize a memristor.
Q2: Why is the memristor not relevant in linear circuit theory?
1. If M(q(t)) is a constant:
2. Principle of superposition is not* applicable:
Slide Number: 8/24

9. Memristive Devices [3] The functions R and f are defined as:
BUT, good news: Chua and Kang extended the definition of a memristor to memristive devices (systems) [3]. Here, the internal state and memristance need not be a function of current alone. Nevertheless, a memristive device is a general memristor (as opposed to charge and flux-controlled memristors that are special memristors). This is because a memristive device’s output voltage is zero, whenever the input current is zero. In other words, the i-v graphs of physical memristive devices are also characterized by pinched-hysteresis loops. Chua and Kang identified practical systems that can be modeled using memristive systems. I am going to use “memristor” for memristive devices from now on (we are going to use a current-controlled one-port memristive device).
Note tha Dr. Jevtic and Dr. Muthuswamy experimentally showed memristive device behavior in the common discharge tube (June 2013).

10. Outline I. Prerequisites for understanding this talk:
1. First course in circuit theory
2. First course in differential equations
II. Background
1. Introduction to Chaos
2. The fundamental circuit elements
III. The Memristor
1. Properties of the memristor
2. Memristive Devices
IV. The Muthuswamy-Chua circuit (System)
1. Derivation of Circuit Equations
2. Hardware emulation of the memristor
3. MultiSim (SPICE) simulation
4. An attractor from the circuit
5. Rigorous Mathematical Analysis
6. A Chaotic Hierarchy
7. Potential conservative chaotic systems
V. Conclusions, Current (future) work and References

11. Derivation of Circuit Equations [6]
System equations:
Specifically:
Parameters:
Using the new memristor, we have the circuit equations shown. Note that I have incorporated the fact that iL = -iM in the memristance (R) and internal state (f). Now, in (7) we are free to pick R(z,y) and f(z,y). Two requirements must be met by R(z,y) and f(y): 1. We should be able to emulate the memristor physically AND 2. We should be able to obtain chaos. Keeping 1. in mind and motivated by the work of Rossler, I picked R(z,y) = R(z) = beta*(z2-1), f(z,y) = -y –alpha*z+y*z. The idea is that the equilibrium point(s) of the system is an unstable saddle-focus. The eq. point of the system is: (0,0,0). Unstable in the y-z plane, hence oscillations start increasing in the y-z plane. Because of the increase in y values, x starts increasing (first equation in (9)). But, because of the increase in x and y, y again starts decreasing (second equation in (9), first terms dominate). These contradicting behaviors should give rise to chaos.

12. Hardware emulation of the Memristor [6]
Bart’s Happy, so yes! Notice that we don’t have an analog computer. Rather, we emulate a memristor as follows. We use sense resistor Rs and difference amplifier U3B to covert current into voltage. Next, we use multipliers U4, U5 and another difference amplifier U3A to obtain vM. Notice that vM is connected to sense resistor Rs, so the KVL equation is actually: vL + vC = vS + vM. The vS causes an offset of 0.01*y when we convert the circuit to dimensionless form.
To realize the internal state, we use op-amp integrator U2B and inverting amplifier U2A. So, the internal state of the memristor is stored in capacitor Cf.
Also, we have scaled all components to realistic values.

13. MultiSim (SPICE simulation)
This was actually achieved last week  while I was at TIFAC-CORE on a nice Tuesday afternoon.

14. An attractor from the circuit [6]
Bart’s happy, so yes again! The attractor on the bottom shows a little distortion.

15. Rigorous Mathematical Analysis
Paper by Ginoux et. al. “Topological Analysis of Chaotic Solution of Three-
Element Memristive Circuit”. International Journal of Bifurcation and Chaos,
Vol. 20, No. 11., pp – 3829, Nov
Already done (?) by for the case in [6] by Ginoux et. al. They compared my attractor to the Rossler attractor, the attractors look very similar because of the way in which I derived the nonlinearities mimicked Rossler’s thinking. They also proved that this system is topologically conjugate to the Rossler system.

16. A Chaotic Hierarchy [5] System 1 phase portrait
Blue: System 1 (differentiable)
Red: System 2 (non-differentiable)
In all cases, system has only ONE equilibrium point at the origin .
Note that delta_a, delta_b, gamma and sigma are selected such that R1=R2 at the turning points. This implies that beta is the only
bifurcation parameter.
Show other results AChaoticHeirarchy pdf – explain why is this a chaotic hierarchy.
I never pursued this further but mention that some authors have actually submitted a publication related to this pdf .

17. Potential Conservative Chaos
BUT, in the manuscript submitted that shows the use of Chebyshev polynomials, the author(s) have not simplified the memristor state function as I have.
KY dimension implies system is conservative BUT has a non-zero divergence (IIT discussion from last week, many thanks). This implies system is not GLOBALLY conservative, so would be interesting to physically implement.

18. Potential Conservative Chaos (contd.)
Now, a live demo of the POTENTIAL
physical implementation…
Simulation results.
Show FPGA realization of Type-3 Order-0 Exponent-2 on FPGA (refer to Valli’s talk for details). Simple Euler’s method.
Applications to Turbulence-modeling via higher Type (dimensional) Muthuswamy-Chua circuit cascades (show Results-February2013.pdf).
Can we obtain hyperchaos in the Muthuswamy-Chua system?

19. Outline I. Prerequisites for understanding this talk:
1. First course in circuit theory
2. First course in differential equations
II. Background
1. Introduction to Chaos
2. The fundamental circuit elements
III. The Memristor
1. Properties of the memristor
2. Memristive Devices
IV. The Muthuswamy-Chua circuit (System)
1. Derivation of Circuit Equations
2. Hardware emulation of the memristor
3. MultiSim (SPICE) simulation
4. An attractor from the circuit
5. Rigorous Mathematical Analysis
6. A Chaotic Hierarchy
7. Potential conservative chaotic systems
V. Conclusions, Current (future) work and References
Talking about conservative chaos leads to conclusions (actually current (future) work) and references.

20. Conclusions and Current (future) Work
I. Conclusions:
1. We obtained a circuit that uses only three fundamental
circuit elements (only one active) to obtain chaos.
2. We can pick our choice of nonlinearity, we discussed a few choices
II. Current (future) work:
1. Work with colleagues at the University of Western Australia (VIT and IIT Chennai, related to the ideal memristor: cos(phi) term in the Josephson Junction) .
2. Is the Muthuswamy-Chua circuit canonical, like Chua’s circuit?
*Utilizing physical memristive devices like discharge tubes. Note that circuit needs a nonlinear, active element
In conclusion, the system we have is shown. We found out that the system can be modeled by three FUNDAMENTAL CIRCUIT elements in series (one active), each element provides one of the three states required for chaos. We can also pick R(z), f(z,y). This talk illustrated one choice.
Future : potential conservative chaotic circuits using a the ideal memristor: Josephson junctions. Also, canonical Muthuswamy-Chua circuit? What about utilizing a physical memristive device?

21. Current (future) Work
Capacitor C models junction capacitance, the flux-controlled memristor M models the interference among quasi-particle pairs, the nonlinear inductor L is the standard junction current. The shunt composed of the linear resistor R and the linear inductor L is used in high frequency applications.

22. Current (future) Work contd.
Plot of (v(t),ix(t)) via a Mathematica simulation. Initial conditions are phi(0)=0,v(0)=1.25, ix(0)=1.4.
Parameters are C=1, i=1, G=1, k0=2,I0=2, L=8,R=1. Simulation was carried out for steps using explicit Euler method.

23. A Canonical Muthuswamy-Chua circuit?
The Muthuswamy-Chua SYSTEM:
Can the system above exhibit canonical behavior like the global unfolding in Chua’s circuit?
Utilizing a Physical Memristive Device like the discharge tube?
Note that system has only one equilibrium point. This was proving to be restrictive but as evident from the chaotic hierarchy, system has “rich” behavior. Hence the first bullet is a very interesting question to pursue.
Also, if we can utilize the discharge tube in chaotic circuits: simple apparatus for demonstrating chaos in (cold) plasmas. Note that we may have to make either the inductor or capacitor “fully” nonlinear (all physical inductors and capacitors have some nonlinearity in them ).

24. References
Alligood, K. T., Sauer, T. and Yorke, J. A. Chaos: An Introduction to Dynamical Systems. Springer, 1997.
Chua, L. O. “Memristor-The Missing Circuit Element”. IEEE Transactions on Circuit Theory, Vol. CT-18, No. 5, pp September 1971.
Chua, L. O. and Kang, S. M. “Memristive Devices and Systems”. Proceedings of the IEEE, Vol. 64, No. 2, pp February 1976.
Hirsch, M. W., Smale S. and Devaney, R. Differential Equations, Dynamical Systems and An Introduction To Chaos. 2nd Edition, Elsevier, 2004.
McCullough, M. C., Iu, H. and Muthuswamy, B. “Chaotic Behaviour in a Three Element Memristor Based Circuit using Fourth Order Polynomial and PWL Nonlinearity”. To appear in the Proceedings of the 2013 IEEE International Symposium on Circuits and Systems.
Muthuswamy, B. and Chua, L. O. “Simplest Chaotic Circuit”. International Journal of Bifurcation and Chaos, vol. 20, No. 5, pp May 2010.
Lorenz, E. N. “Deterministic Nonperiodic Flow”. Journal of Atmospheric Sciences, vol. 20, pp/ , 1963.
Sprott, J. C. “Simple Chaotic Systems and Circuits”. American Journal of Physics, vol. 68, pp , 2000.
Strogatz, S. H. Nonlinear Dynamics and Chaos. Addison-Wesley, 1994.
Strukov, D. B., Snider, G. S., Steward, D. R. and Williams, S. R. “The missing memristor found”. Nature, vol. 453, pp , 1st May 2008.
Questions welcome.
Questions?