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Phase Plots in Dynamical Systems • January 2015 • Chaos Theory Phase Plots in Dynamical Systems 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 1 Phase Plots in Dynamical Systems • January 2015 • Chaos Theory 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 equa- tion 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¨ − m(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 m = 1/2 and figure 2 with For such a differential equation an equiva- m = 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 condi- tions. 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) = −m(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 (m = 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 cy- cle is stable. As can be seen for the Van der Pol oscilla- tor with F(x) = −m(x − x3/3) with m 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 (m = 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 equa- tion. 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 re- duced 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 + m(1 − x ). This results in figures 3 and 4. How does the path in phase space de- 2 x¨ − m(1 − x )x˙ + x = A sin(wt + q0) (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.
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