Phase Plots in Dynamical Systems • January 2015 • Chaos Theory Phase Plots in Dynamical Systems

Jolien L. Diekema,Folkert S.J. Nobels and Eifion H. Prinsen University of Groningen [email protected] [email protected] [email protected]

Abstract

In this essay the properties of phase plots of different dynamical systems will be investigate. Phase plots are a useful tool to get more insight in the development of a system once the initial conditions are known. First the general method for simpler differential equations will be discussed.We will take a look at more complicated systems, such as the van der pol-duffing oscillator and investigate limit cycles and slope fields. The Van der Pol oscillator exhibits chaotic behavior when a driving force with sufficient amplitude is added.

1. Introduction tigated after which a closer look will be taken at phase plots of more complicated dynamical systems. Dynamical systems are mechanisms that de- velop in time deterministically. Examples in- clude models for the flow of water in a pipe 2. Phase Plots of Differential or the amount of fish in a lake every spring. Equation without Dependence in x˙ [1] Dynamic systems evolve deterministically, meaning their future behavior is determined In many areas of physics differential equations by their initial conditions. However, small like equation 1 are found. For example the changes in the initial conditions can make a differential equation of the harmonic oscilla- great difference in the evolution of the system. tor and the differential equation of an ideal Therefore, simple nonlinear dynamical systems pendulum are both of this category of dif- can exhibit completely unpredictable behavior, ferential equations. Although these differen- in spite of being fundamentally deterministic. tial equations are not always analytically solv- This is called deterministic chaos. With the in- able, an equation describing the lines in phase vestigation of dynamical system, the focus lies space can be determined analytically without not on solving the equations describing the sys- approximations[2]. tem exactly, because this can be really hard for more complicated system. But instead the fo- d2x = F(x) (1) cus lies on analyzing the systems to investigate dt2 whether the system evolves to a steady state or if the long term behavior depends on the initial Every differential equation of the form of equa- conditions for example. To obtain insight in tion 1 can be written as a coupled system of these systems, phase plots are a very useful differential equations. This can be done by tool. In phase space, every degree of freedom writing a separate differential equation for the of a system is represented as an axis. Because velocity. So if this is done by equation 1, equa- every possible state of a system, or the system’s tions 2 and 3 are obtained[2]. evolving trajectory, completely depends on the initial conditions, a phase space diagram rep- dx resents all states a system with fixed initial = y (2) dt conditions can be in. In this paper, phase plots dy of easily exactly solvable systems will be inves- = F(x) (3) dt

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These equations can be combined to cancel out of differential equations in phase space. As the time dependence and reducing the system long as this equation is integrable the equa- to one differential equation. For this step equation provides information about the behavior tion 4 is used. After using this trick we obtain in phase space. equation 5[2]. Z y 0 Z x y 0 0 0 dy 0 dy = f (x )dx (9) dy y0 g(y ) x0 = dt (4) dx dx dt 3.2. Non-separable differential equa- dy F(x) = (5) tions dx y Another common type of differential equations As can be seen this equation is separable and with dependence in x˙ is an equation of the thus we can obtain the equation describing the form behavior of a differential equation like equation d2x dx 1 in phase space. The equation describing the = F(x) − c (10) dt2 dt behavior in phase space is given by equation Using the same method as before, it can be 6[2]. reduced to one differential equation 11. Z x 1 2 0 0 y = F(x )dx (6) dy F(x) 2 x0 = − c (11) dx y Equation 6 can be rewritten to show that we have an equation which implies energy conser- This differential equation is not separable; vation; see equation 7[2]. it is not possible to find an implicit or explicit equation for y. However, despite this an idea 1 Z x for the solution can still be obtained by mak- y2 − F(x0)dx0 = 0 (7) 2 x0 ing a slope field of the equation. This will be |{z} | {z } explained in the next section, applying it to the Kinetic E Potential E Van der Pol equation. This means that for differential equations of the form of 1 the equations describing phase space can be found if the function F(x) is integrable. 4. Van der Pol Equation The Van der Pol equation is a nonlinear oscil- 3. Phase Plots of Differential lator, which has its origin in electrodynamics. Equations with Dependence in x˙ Energy is dissipated at high amplitudes but supplied at small amplitudes. [3, 1, 4]. Because 3.1. Separable Differential Equations of this, there exists a state where there is a balance between the removed and added en- The previous differential equation did not de- ergy. This state, towards which the oscillations pend on y. It is obvious that there are more converge, is known as a limit cycle and is inde- complicated differential equations with a de- pendent of the initial conditions. [4]. A limit pendence in x˙, that have the form of equation cycle is a closed trajectory in phase space hav- 8. d2x ing the property that other trajectories spiral = F(x, y) = f (x)g(y) (8) dt2 into it if time approaches infinity. Although slightly more complicated than the x¨ − µ(1 − x2)x˙ + x = 0 (12) previous equation, this equation can be solved as well using the same tricks as with the pre- Using the van der Pol equation, a simulation vious differential equation. When this is done, showing the development of the path in phase equation 9 describes the behavior of this type space for 7 randomly chosen points was made,

2 Phase Plots in Dynamical Systems • January 2015 • Chaos Theory see figure 1 with µ = 1/2 and figure 2 with For such a differential equation an equiva- µ = 4. As can be seen in both of these figures, lent system can be formed, which is given by the Van der Pol oscillator seems to have limit equations 14[5]. cycle that doesn’t depend on the initial conditions. This raises the question: Does the Van dx der Pol oscillator always have a limit cycle, and = y − F(x) (14a) dt if so, why? dy = −G0(x) (14b) dt

As can be seen the Van der Pol equation (equation 12) is a Lienard system with the F(x) = −µ(x − x3/3) and G(x) = x2/2[5, 4]. This is due to the Van der Pol equation being a Lienard system, one can look if it is possible then to apply Lienard theorem.

4.1.1 Lienard Theorem Lienard theorem is a theorem which applies for Lienard systems if the following conditions are satisﬁed Figure 1: As can be seen for 7 random points, the Van der Pol oscillator has one limit cycle, indepen- • F(x) and G0(x) are smooth odd functions dent of the initial conditions (µ = 0.5). • G0(x) > 0 for x > 0

• F(x) has exactly 3 zeros -a, 0 and a.

• F0(0) < 0

• F(x) ≥ 0 for x > a

• F(x) → ∞ for x → ∞

If these conditions are satisﬁed, the Lienard system has exactly one limit cycle and this cycle is stable. As can be seen for the Van der Pol oscillator with F(x) = −µ(x − x3/3) with µ positive and G0(x) = x, this function exactly matches the criteria for Lienard Theorem. This means Figure 2: As can be seen for 7 random point, the van der that a non-driven Van der Pol equation has a pol oscillator has one limit cycle, independent unique limit cycle, independent of the initial of the initial conditions (µ = 4). conditions.

4.1. Lienard Systems & Limit Cycles 4.2. Slope Field A Lienard system is a differential equation of A slope ﬁeld is a graphical representation of the form of equation 13[5]. the solutions of a ﬁrst order differential equation. This is useful because it can be created d2x dx + F0(x) + G0(x) = 0 (13) without solving the equation analytically and dt2 dt

3 Phase Plots in Dynamical Systems • January 2015 • Chaos Theory gives an indication of what solutions look like. 5. Driven Van der Pol Oscillator The second order differential equations are reduced to first order using equation 11. For The driven Van der Pol oscillator is given by the Van der Pol equation this means that equation 15. dy −x 2 dx = y + µ(1 − x ). This results in figures 3 and 4. How does the path in phase space de- 2 x¨ − µ(1 − x )x˙ + x = A sin(ωt + θ0) (15) velop in such a slope field? The path taken in phase space is the path that follows the direc- In this equation there exist two frequencies tion of the lines, this means for the slope plots (one of the driving force and the original sys- in figure 3 and 4 that the trajectories evolve in tem), so this system can be treated as a coupled the same way as the one in the situation with dynamic system. In the driven Van der Pol os- the 7 random points, given in figure 1 and 2. cillator, some interesting behavior appears. To explain this, we begin with a short introduction to resonance and Arnold tongues.

5.1. Resonance and Arnold Tongues

A driven and damped pendulum is a coupling of two oscillators with two different frequencies. To look at what happens when two oscillators are coupled together we introduce a coupled system for which α is the rotation frequency and β is the amplitude of the driving p force. For a rational α (= q , p, q ∈ N), q itera- Figure 3: Slope field of the Van der Pol equation, hori- tions will return the system to its starting value zontal x and vertical y (µ = 0.5). after making p rounds of the unit interval and we get a periodic orbit. But the driving force (with amplitude β) gives some additional oscil- latory contribution and non-linearly modulates the rotation. The nonlinear nature of this per- turbation provides room to absorb the ’misfit’ and maintain the periodic motion even for an irrational α, as α is varied continuously over a finite interval. We say that the oscillations are phase-locked or in resonance. The frequency of the pendulum is entrained to a rational multiple of the driving frequency. It is a balance between the system trying to oscillate at its natural frequency as well as at the frequency of the external drive. Any quasi-periodic motion is the result of not being able to reach this balance. As we vary the frequency parameter Figure 4: Slope field of the Van der Pol equation, hori- α for a fixed amplitude β we pass through both zontal x and vertical y (µ = 4). the periodic and the quasi-periodic motions. A plot of the parameter space of such a system[6] looks like:

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der Pol oscillator resonates. The behavior of the driven van der Pol equation depends on the magnitude of the amplitude of the driving force. For a very small amplitude A 1, the free movement dominates and the driving force only leads to small perturbations. It is only at bigger amplitudes that chaos and phase-locking becomes into play. For a large amplitude, as before, the system resonates at the driving frequency. [8]

Figure 5: Arnold Tongue Circle Map [6]

Regions called ’Arnold tongues’ appear 6. Van der Pol-Duffing equation in parameter space, and are shown as black regions in the picture. This is where the two oscillators mode-lock and their motion is pe- The Van der Pol-Dufﬁng equation is a non- riodic with a common frequency. Outside linear differential equation similar to the Van these regions, this does not happen and their der Pol equation but with the addition of an combined motion has two independent fre- extra term proportional to x3. The Van der Pol- quencies. Dufﬁng equation is given by equation 16. In the case of drive Van der Pol equation there exist two frequencies, the frequency of self- oscillation determined by µ and the frequency of the periodic forcing. If the parameters are inside the tongue, these frequencies will res- x¨ − µ(1 − x2)x˙ + x + βx3 = 0 (16) onate at a frequency that is an integer multiple of the driving frequency.

6.1. Limit Cycle

Like the Van der Pol equation, the Van der Pol-Duffing equation is a Lienard system as well: The Lienard theorem can be applied to the Van der Pol-Duffing equation with F(x) = −µ(x − x3/3) and G0(x) = x + βx3. This means that the Van der Pol-Duffing oscillator also has a unique limit cycle, independent Figure 6: Arnold tongues[7] of the initial conditions. Plots were made for the phase space evolution of the Van der Pol- As can be seen in this picture, there are Duffing equation for 2 conditions, see figure various zones of parameters for which the Van 8

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8. Bifurcation Theory

A bifurcation of a dynamical system is a quali- tative (topological) change in its behavior due to variation of one of the parameters. There are many types of bifurcations but a particularly interesting type in the case of the Van der Pol oscillator is Saddle-Node Bifurcation. A typical example of a one-dimensional differential equation with a Saddle-Node Bifurcation is:

Figure 7: As can be seen for 7 random points, the Van 2 der Pol-Dufﬁng oscillator has one limit cycle, x˙ = λ + x , λ ∈ R (18) independent of the initial conditions (µ = 0.5 and β = 0.01). Depending on the value of λ solutions can have 2, 1 or 0 equilibria: For λ < 0 there are√ 2 equilibria, a stable equilibrium at x =√− −λ and an unstable equilibrium at x = −λ;

For λ > 0 there are no equilibria;

And for λ = 0 there is one critical equilibrium point; a saddle-node point, sometimes called the bifurcation point.

A detailed analysis of bifurcation of the Van der Pol equation is beyond the scope of this paper. However it is important to realize that it is applicable to analyze the chaotic behavior Figure 8: For 7 random point, the Van der Pol-Dufﬁng of the Van der Pol equation and the Van der oscillator has one limit cycle, independent of Pol-Dufﬁng equation. the initial conditions (µ = 4 and β = 0.01).

7. Driven Van der Pol-Duffing 9. Poincaré-Bendixson Theory Equation Another interesting theory that is applicable to Similar to the Van der Pol equation, the Van the discussed systems is Poincaré-Bendixson der Pol-Dufﬁng equation can be driven by a Theory. It states that when certain continu- periodic force. The driven Van der Pol-dufﬁng ity conditions are met, a trajectory of a system equation is given by equation 17. must be either a closed orbit, approach a closed orbit or approach an equilibrium when t → ∞. This theory however is beyond the scope of 2 3 x¨ − µ(1 − x )x˙ + x + βx = A sin(ωt + θ0) this paper as well. It is still worth mentioning (17) however.

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10. Conclusion [3] E. W. Weisstein. Van der pol equation. math- Both the Van der Pol equation and the Van world.wolfram.com/vanderPolEquation.html, der Pol-Duffing equation are Liénard systems. Dec. 2014. This means that they possess one unique and stable limit cycle. This limit cycle is indepen- [4] W. Cao. Van der pol oscilla- dent of the initial conditions. The slope fields tor. www.math.cornell.edu/∼ give insight in the solutions of systems without templier/junior/final_paper/Wesley_Cao- actually solving them. vanderpol.pdf, 2013. By analyzing the Arnold tongues and using [5] O. Knill. Lienhard systems. bifurcation theory the chaotic behavior of the www.math.harvard.edu/archive/118r_spring- systems can be investigated. For the driven _05/handouts/lienard.pdf, Dec. 2014. Van der Pol oscillator shows chaotic behavior if the amplitude of the driving force is large [6] N. Kumar. Deterministic chaos, complex enough. change out of simple necessity, 1996.

[7] Oscillators of van der pol References and van der pol-dufﬁng. http://www.sgtnd.narod.ru/science/atlas- [1] H. Broer and F. Takens. Dynamical systems /eng/charts/vdpd.htm. assessed: 08-01- and chaos. Springer, 2011. 2015.

[2] H. Broer, J. Epema, and M. Kuipers. [8] U Parlitz and W Lauterborn. Period- Trillingen en slingerbewegingen vanuit doubling cascades and devil’s staircases wiskundig oogpunt, 1987. of the driven van der pol oscillator.

11. Appendix:Numerical Approximation

A lot of differential equations can be approximated using numerical methods, the numerical method which is used in this paper is based on approximating the derivatives with three point midpoint equations. For the ﬁrst derivative this is equation 19 and for the second derivative this becomes equation 20.

x[t + δ] − x[t − δ] x˙[t] = (19) 2δ

x[t + δ] − 2x[t] + x[t − δ] x¨[t] = (20) δ2 With the help of these approximations ﬁrst the numerical approximation of the Van der Pol equation is calculated after which the approximation to the Van der Pol-Dufﬁng oscillator is calculated.

11.1. Van der Pol equation

The Van der Pol equation can be numerical approximated, with the help of the equation of the previous set of equation (19 and 20). The Van der Pol equation is given by equation 21.

x¨ − µ(1 − x2)x˙ + x = F(t) (21)

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Which becomes

x[t + δ] 2x[t] x[t − δ] µ(1 − x[t]2)x[t + δ] µ(1 − x[t]2)x[t − δ] − + − + + x[t] = F(t) (22) δ2 δ2 δ2 2δ 2δ After reordering, rewriting and multiplying the fraction with δ2/δ2 equation 23 for x[t + δ] is obtained.

2 2 µδ 2 δ F(t) + 2 − δ x[t] − x[t − δ] 1 + 2 1 − x[t] x[t + δ] = (23) µδ 2 1 − 2 1 − x[t]

11.2. Van der Pol-Dufﬁng In a similar way the Van der Pol-dufﬁng equation can be numerically approximated, this results in equation 24

2 3 2 2 µδ 2 δ F(t) − x[t] δ + 2 − δ x[t] − x[t − δ] 1 + 2 1 − x[t] x[t + δ] = (24) µδ 2 1 − 2 1 − x[t]