The equation of motion of a dynamical system is given by a set of differential equations. That is (1)

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1 Dynamcal Systems Many engneerng and natural systems are dynamcal systems. For example a pendulum s a dynamcal system. State l The state of the dynamcal system specfes t condtons. For a pendulum n the absence of external exctaton shown n the fgure, the angle θ and the angular velocty θ & unquely defne the state of the dynamcal system. Phase Space mg Fgure 1. Smple pendulum. Plots of the state varables aganst one another are referred to as the phase space representaton. Every pont n the phase space dentfes a unque state of the system. For the pendulum, a plot of θ & versus θ s the phase space representaton. Equaton of Moton The equaton of moton of a dynamcal system s gven by a set of dfferental equatons. That s x& = f(x, t) (1) where x s the state and t s tme. The dynamcal system s lnear f the governng equaton s lnear. For the pendulum shown n Fgure 1, the equaton of moton s gven as θ & = ω ω & = ζωoω ω o sn θ where ω = g o l () and the dynamcal system s nonlnear. For small ampltude oscllaton, the equaton of moton becomes θ & = ω ω & = ζωoω ωoθ sn θ θ, and (3) The dynamcal system s now lnear. In Equatons () and (3) ωo s the natural frequency and ζ s the dampng coeffcent. Autonomous and Nonautonomous Systems A system s sad to autonomous f tme does not appear explctly n the equaton of moton. The equaton of moton of nonautonomous systems, however, explctly 1

2 depends on tme. Thus, the equatons of moton gven by () and (3) for a pendulum n absence of external exctaton are for autonomous systems, whle a forced pendulum s a nonautonomous system. A nonautonomous system could be determnstc or stochastc. Orbt An orbt or trajectory s a curve n phase space, whch s obtaned by the soluton of the equaton of moton. Flow For a fxed tme, f(x,t), the rght-hand sde of the equaton of moton gven by (1), dentfes a vector feld n the phase space that s tangent to the trajectores. Fgure shows the phase space and the flow for a damped pendulum gven by Equaton (). The black arrows of the vector feld f = ( ω, ζω oω ωo sn θ) are tangental to all trajectores n the phase space. A two-dmensonal phase space provdes a qualtatve pcture of the behavor of the dynamcal systems ncludng ts orbt and ts flow. In Fgure the dashed red lnes are null clnes. These are the lnes where the tme dervatve of one component of the state varable s zero. For the damped pendulum t follows that θ & = ω = 0 ω = ωo sn θ / ζ (3) Therefore, one null clne s the angle θ axes and the other s a snusodal curve n the phase space. On the frst null clne the flow vector feld s vertcal (snce θ & = 0 ), whle on the second one, the vector feld s horzontal (snce ω& = 0 ). Fgure. Schematcs of the phase plane for the damped pendulum. (

3 Between the null clnes the drecton of the flow vector s determned by the sgns of θ & and ω. & At the ntersecton ponts of the null clnes, θ & =ω= & 0, and the flow vector feld s zero. These ntersecton ponts, whch correspond to the statonary solutons, are called fxed ponts or equlbrum ponts. Fxed ponts of a dynamcal system could be stable or unstable. Poncaré Secton and Poncaré Map Poncaré secton and Poncaré map are tools for vsualzaton of the flow n a phase space. Poncaré secton s a plane (or curved surface) n the phase space that s crossed by almost all orbts. The Poncaré map maps the ponts of the Poncaré secton onto tself. Consecutve ntersecton ponts of the orbt wth the Poncaré secton form the Poncaré map. A Poncaré map represents a contnuous dynamcal system by a dscrete one. Poncaré maps are nvertable maps because one gets from by followng the orbt u n u n + 1 Fgure 3. Schematcs of Poncaré secton and Poncaré map. backwards. For the case of the forced pendulum the Poncaré secton s defned by a certan phase of the tme-perodc exctaton. Non-Wanderng Set Non-wanderng set s a set of ponts n the phase space for whch all orbts startng from a pont of ths set come arbtrarly close and arbtrarly often to any pont of the set. Non-wanderng sets are of four types. These are Fxed (statonary) ponts. For the smple pendulum gven by Equaton (), ω = θ & = 0 and θ = 0 and θ = 180 are fxed ponts. Lmt cycles (perodc solutons). These solutons are common for the lnearzed oscllaton of a smple pendulum. Quas-perodc orbts. Perodc solutons wth at least two ncommensurable frequences. These solutons occur for an undamped pendulum under perodc exctatons. Chaotc orbts. Bound non-perodc solutons. These solutons occur for a (nonlnear) pendulum under certan external exctatons. The frst three types can also occur n lnear dynamcal systems. The fourth type appears only n nonlnear systems. The Poncaré map for lmt cycles become fxed ponts. A non-wanderng set can be ether stable or unstable. Changng a parameter of the system can change the stablty of a non-wanderng set. Ths s accompaned by a change of the number of non-wanderng sets due to a bfurcaton. 3

4 Stablty and Bfurcaton In nonlnear dynamcal system, the man questons are: What s the qualtatve behavor of the dynamcal system? How many non-wanderng sets (fxed ponts, lmt cycles, quas-perodc or chaotc orbts) occur? Whch of the non-wanderng sets are stable? How does the number of non-wanderng sets change wth changes n the parameters of the system? The appearance and dsappearance of non-wanderng sets s called a bfurcaton. Change of stablty and bfurcaton always concde. Stablty A non-wanderng set may be stable or unstable. The stablty could be n the sense of Lyapunov (weak) or Asymptotc (strong). Lyapunov (Margnal) Stablty A non-wanderng set s sad to be Lyapunov stable f every orbt startng n ts neghborhood remans n ts neghborhood. Asymptotc Stablty A non-wanderng set s sad to be asymptotcally stable f n addton to the Lyapunov stablty, every orbt n ts neghborhood approaches the non-wanderng set asymptotcally. Thus, a non-wanderng set s ether asymptotcally stable, margnally stable (Lyapunov stable), or unstable. Attractors and Basn of Attracton Asymptotcally stable non-wanderng sets are also called attractors. The basn of attracton s the set of all ntal states approachng the attractor n the long tme lmt. Lnear Stablty of Non-Wanderng Set To check the stablty of a non-wanderng set a lnear stablty analyss may be used. Let (t) be an orbt that satsfes Equaton (1). That s x o x & f(x t) (4) o = o, The soluton x o (t) s asymptotcally stable f any nfntesmal small perturbaton x(t) decays. Assume that x = xo ( t) + x s the perturbed soluton, and x satsfes the equaton of moton gven by (1). It then follows that 4

5 d( x) dt ( xo ) = x (5) The lnearzed equaton of moton for x(t) gven by (5) s justfed as long as the orbt s n the neghborhood of x o. For fxed ponts, the fundamental solutons for x(t) are x = e λst x s (6) d f where λ s and xs are the egenvalues and egenvector of the Jacoban, and s s the dmenson of phase space. The egenvalues λ s are the roots of the characterstc polynomal The fxed pont del λi =0 (6) x o s asymptotcally stable f the real parts of all egenvalues λ s are negatve. If at least one egenvalue has a postve real part the correspondng fundamental soluton would ncrease exponentally, and the fxed pont wll be unstable. Routh and Hurwtz theorem may be used to check the stablty wthout explctely calculatng the egenvalues. For the case of a two-dmensonal phase space the characterstc polynomal s quadratc. Routh and Hurwtz theorem mples that both egenvalues have negatve real part f and only f det s postve and the trace s negatve. The regon of stablty and nstablty are shown n Fgure 4. Nodes, Spral and Saddle Fgure 4 shows a classfcaton scheme of the fxed (statonary) ponts n twodmensonal phase spaces. The noton spral and node are nspred by the flow near the fxed pont. A par of conjugated complex egenvalues lead to a spral, whereas a node s causes by two real egenvalues of the same sgn. Real egenvalues of dfferent sgn lead to a saddle. That s, a saddle s a fxed pont where at least one egenvalue has a postve real part but also at least one egenvalue has a negatve real part. Near a saddle, an orbt s usually attracted at frst but repelled later on. There are ponts n the phase space that approach the fxed pont as t, and they form the stable manfold. Saddles and ther stable manfolds are usually the boundares of a basn of attracton. The pendulum shown n Fgure 1 has a saddle pont (correspondng to the upsde-down equlbrum poston) and a spral f ζ < 1 or a node f ζ > 1 as shown n Fgure. j 5

6 det j Fgure 4. The regon of stablty and nstablty and classfcaton scheme of the fxed (statonary) ponts n two-dmensonal phase spaces. 6

7 Bfurcaton The number of attractors n a nonlnear dynamcal system can change when a system parameter s vared. Ths change s called bfurcaton and t s accompaned by a change of the stablty of an attractor. In a bfurcaton pont, at least one egenvalue of the Jacoban wll attan a zero real part. There are three generc types of bfurcaton. Statonary Bfurcaton In a statonary bfurcaton, a sngle real egenvalue crosses the boundary of stablty. Hopf Bfurcaton Hopf bfurcaton occurs when a conjugated complex par crosses the boundary of stablty. In the tme-contnuous case, a lmt cycle bfurcates. It has an angular frequency that s gven by the magnary part of the crossng par. In ths secton the man forms one-dmensonal bfurcaton are llustrated. The phase space varable s u. The control parameter s µ. The bfurcaton pont s at µ = 0. The drecton of moton n the one-dmensonal phase space s shown by arrows. Stable (unstable) fxed ponts are drawn as red sold (dotted) lnes. Ptchfork Bfurcaton Ptchfork bfurcaton s possble n dynamcal systems wth an nverson or reflecton symmetry. That s, an equaton of moton that remans unchanged f one changes the sgn of all phase space varables. An example s a system governed by u& = ( µ u )u (7) As the control parameter µ vares the statonary solutons and ther stablty condtons changes. For µ < 0, there s one equlbrum soluton, u=0, whch s stable. For µ > 0, there are three equlbrum solutons u=0, and u = ± µ. Here, u=0 s unstable whle u = ± µ are stable. At µ = 0, a bfurcaton occurs whch s referred to as (supercrtcal) ptchfork bfurcaton. Smlarly for a system governed by 7

8 Fgure 5. Schematcs of a ptchfork bfurcaton. u& = ( µ + u )u (8) there are three equlbrum solutons u=0, and u = ± µ for µ < 0. In ths case u=0 s a stable soluton whle u = ± µ are unstable. For µ > 0, there s only one soluton, u=0, whch s stable. In ths case at µ = 0, a (subercrtcal) ptchfork bfurcaton occurs. Transcrtcal Bfurcaton Consder a dynamcal system gven by u& = ( µ u)u (9) For µ < 0, there are two equlbrum soluton, u=0, whch s stable, and u = µ, whch s unstable. For µ > 0, there are stll the same two equlbrum soluton. But u=0 s unstable, and u = µ s now stable. That s at µ = 0, the two statonary soluton exchanged ther stablty. In ths case t s sad that a transcrtcal bfurcaton occurred. 8