C. Non-linear Difference and Differential Equations: Linearization and Phase Diagram Technique

Transcription

1 C. Non-linear Difference and Differential Equations: Linearization and Phase Diaram Technique So far we have discussed methods of solvin linear difference and differential equations. Let us now discuss the case of nonlinear difference and differential equations. The first point to be noted here is that it is etremel difficult to derive an eact solution to a non-linear difference or differential equation. However, two techniques are often used to draw some qualitative inference about the behaviour of the dnamic sstem: one of these is the linearization technique, and the other is the phase diaram technique. C. Linearization of non-linear difference/differential equations and local stabilit analsis: (a) Sinle non-linear equation: Consider an nonlinear function of a sinle variable : f ( ) : D R where D and R are the domain and rane of the function respectivel. Let ˆ D be some iven value of the variable. Then b Talor s Theorem, the function f () can be epanded around ˆ in the followin wa: f ʹ ( ˆ) f ʹ ( ˆ) f ʹ ʹ ʹ ( ˆ) 3 f ( ) = f ( ˆ) + ( ˆ) + ( ˆ) + ( ˆ) +...!! 3! Now a linear approimation of the non-linear function f () around the point ˆ is iven b: f ʹ ( ˆ) f ( ) f ( ˆ) + ( ˆ)!. i.e., f ( ) f ʹ ( ˆ) + [ f ( ˆ) f ʹ ( ˆ) ˆ ] () Note the linear function iven above is onl an approimation of the f () function around ˆ, i.e., it resembles the f () function onl in a small neihbourhood of ˆ. In eneral this linear function does not closel approimate the f () function for all values of. Thus whatever conclusion we draw on the basis of this linear approimation will onl be valid locall around ˆ. Linearization technique is often used to convert a non-linear difference or differential equation into a linear form. Generall the non-linear equation is linearl approimated around its stead state value. This allows us to derive some conclusions about the time path of the variable in the neihbourhood of the stead state and thus its local stabilit propert. First let us consider a non-linear difference equation of the form: t = f ( t ) () (We are inorin the parameter vector α for the time bein). We know that the stead state of the above difference equation is defined as =, i.e., = f (). Suppose there is indeed a that solves this equation. In t t = other words, suppose a stead state eists. Then linearizin the f ( t ) equation around, we et a linear differential equation of the form: 3

2 t f ( ) f ( ) f ʹ t + ( )( t ) f ʹ ( ) t + [ f ( ) f ʹ ( ) ] = = = a t + b (ʹ ) t Solution: t = C( a) +, C an arbitrar constant. Stabilit depends on the term a, i.e., on the term f ʹ (). If f ʹ ( ) < the sstem is locall stable; if f ʹ ( ) > the sstem is locall unstable. We can proceed to analse the local stabilit propert of a non-linear differential equation in an analoous manner. Consider a non-linear differential equation of the form: d = f () (3) The stead state of this differential equation is defined b d = 0, i.e., f ( ) = 0. Suppose there is a that satisfies this stead state condition. Then linearizin f () around, we can write the differential equation (3) as: d f ( ) + f ʹ ( )( ) = f ʹ ( ) f ʹ ( ) [since f ( ) = 0 ] = a b (3ʹ ) Solution: ( t) = Cep + ; C an arbitrar constant. Once aain stabilit depend on term a, i.e., on f ʹ (). If f ʹ ( ) < 0, the sstem is locall stable; if f ʹ ( ) > 0, the sstem is locall unstable. (b) dimensional sstem of non-linear equations: Let us first consider the followin non-linear sstem of differential equations: d = f (, d = (, (4) The stead state of this sstem is defined as (, such that f (, = 0 and (, = 0. Suppose that a stead state eists. Then for the two variable functions f (, and (,, we can derive the linear approimations around (, as: f (, f (, + f (, ( ) + f (, ( (, (, + (, ( ) + (, ( Notin that f (, = 0 and (, = 0 (from the definition of the stead state) and simplifin, we can transform the non-linear sstem iven in (4) to a correspondin linear sstem of the followin form: 4

3 d = f (, + f (, [ f (, + f (, ] = a + a b (4ʹ ) d = (, + (, [ (, + (, ] = a + a b The coefficient matri of this linear sstem is iven b the followin Jacobian matri of partial derivatives of f (, and (, (with the derivatives bein evaluated at the stead state): f (, f (, A = (5) (, (, If the eplicit function forms of f (, and (, were known to us, then we could directl calculate the stead state values of and and find out the eact numerical values of these partial derivatives evaluated at that stead state. Thus we could then find the eact form of the co-efficient matri and proceed to solve the sstem of linear differential equations iven in (4ʹ ) followin the method discussed earlier. But even when the eact functional forms of f (, and (, are not known, we can still sa somethin about the stabilit of the sstem without actuall solvin the sstem, provided we are iven certain information about these partial derivatives. To see how, first recall that for a sstem of linear differential equations, stabilit of the sstem depends cruciall on the characteristics roots of the co-efficient matri. If the co-efficient matri has real characteristic roots (either distinct or repeated), then the dnamic sstem is stable if the roots are neative; is unstable if the roots are positive, and is a saddle point if one root is positive and one is neative. On the other hand, if the coefficient matri has comple characteristic roots of the eneral form u ± iv, then the stabilit depends on the real part of the comple root u: the sstem is oscillator stable (ccles with decreasin amplitudes) if u < 0 ; the sstem is oscillator unstable (ccles with increasin amplitudes) if u > 0 ; and the sstem will show uniform ccles with constant amplitudes (neither stable nor unstable) if u = 0. Thus in order to be able to sa somethin about the (local) stabilit of sstem iven in (4ʹ ), we have to eamine the characteristics roots of the co-efficient matri iven in (5). Note that the characteristic equation for the Jacobian matri iven in (5) is: f (, λ f (, = 0 (, (, λ i.e., [ f (, λ ] [ (, λ] f (,. (, = 0 Simplifin, λ [ f (, + (, ] + [ f (,. (, f (,. (, ] = 0 i.e., λ [TraceA] λ + DetA = 0 (6) Now form the theories of quadratic equations, we know that for an quadratic equation of the form p + q = 0 has two roots and such that + = p and. = q. Applin this theorem to (6) (which is a quadratic equation in λ ), we know that the two characteristics roots would be such that λ + λ = TraceA. λ = DetA. We can now proceed to discuss different alternative (and mutuall eclusive) possibilities: 5

4 CASE I: (TraceA) 4DetA 0 (implin that the roots are real) Subcase (a): DetA<0; TraceA could be anthin (i.e., reater than, equal to, or less than zero). Note that DetA<0 implies λ λ < 0, i.e., λ are have opposite sins. Thus one of them would be positive and one neative. Hence in this case the equilibrium will be locall a saddle point. Subcase (b): DetA>0; TraceA>0. In this case DetA>0 implin λ λ > 0, i.e., λ are of same sin either both are positive or both are neative. But it also iven that TraceA>0 implin λ + λ > 0. Thus both λ must be positive. Hence in this case the equilibrium is locall unstable. Subcase (c): DetA>0; TraceA<0. In this case once aain DetA = λ λ > 0; hence λ have the same sins. But now TraceA<0 implin λ + λ < 0. Thus both λ must be neative. Hence in this case the equilibrium is locall stable. Remark. The determinant of the co-efficient matri can never be zero for an linearl independent sstem. Thus DetA is either positive or neative. Remark. Note that when the roots are real (i.e., (TraceA) 4DetA 0), the possibilit that DetA>0 and TraceA=0 cannot arise because when TraceA=0, then the roots will be real if and onl if 4 DetA > 0 DetA < 0. CASE II: (TraceA) 4DetA < 0 (implin that the roots are comple) In this case, the roots will have the eneral form a ± ib. Therefore, Trace A= λ+ λ = ( a+ ib) + ( a ib) = a Det A= λλ = ( a+ ib)( a ib) = a + b Now recall that for a two dimensional sstem of linear differential equations with comple roots, the stabilit of the sstem depends on the real part of the comple root a. Therefore, in this case, we can determine the stabilit propert of the sstem b simpl lookin the TraceA. TraceA> 0 a > 0; hence the sstem will be unstable (unstable oscillations) TraceA< 0 a< 0 ; hence the sstem will be stable (stable oscillations) TraceA= 0 a = 0 ; hence the sstem will ehibit uniform ccles which are neither stable nor unstable. Let us net consider a non-linear sstem of difference equations, iven b t = f ( t, t ) (7) t = ( t, t ) The stead state of this sstem of difference equations is defined as (, such that = f (, and = (,. Suppose that a stead state eists. Then for the two 6

5 variable functions f ( t, t ) and ( t, t ), we can derive the linear approimations around (, as: f, ) f (, + f (, ( ) + f (, ( ) ( t t t t ( t, t ) (, + (, ( t ) + (, ( t Usin these two linear approimations, we can transform the nonlinear sstem of difference equation iven in (7) to the followin linear sstem: t = f (, t + f (, t + [ f (, f (, + f (, ] (7ʹ ) t = (, t + (, t + [ (, (, + (, ] Like the differential equation case, the coefficient matri of this linear sstem is aain iven b the Jacobian matri: f (, f (, A = (, (, We know that the stabilit of the sstem once aain depends on the characteristic roots of this co-efficient matri. However unlike the differential equation case, if the roots are real, stabilit condition now requires that the absolute values of both these roots be less than unit. If the roots are real and have absolute values reater than unit, the sstem in unstable. If one of the real roots have absolute value reater than unit and the other one has absolute value less than unit, then equilibrium is a saddle point. Also recall that when the roots are comple and of the eneral form u ± iv, then stabilit depend on the modulus r = u + v. The sstem will show stable oscillations if r < ; will show unstable oscillations if r > ; and will be characterised b uniform oscillations (neither stable nor unstable) if r =. Given the co-efficient matri A, we can aain derive the characteristic equation as λ [ f (, + (, ] + [ f (,. (, f (,. (, ] = 0 i.e., λ [TraceA] λ + DetA = 0. As before, the characteristic roots would be such that λ + λ = TraceA and λ. λ = DetA. Thus as we did in the differential equation case, b eaminin the sins of TraceA and DetA, we can derive some conclusions about the sins of the characteristics roots. However that information is now not sufficient for stabilit. In order to be able to sa somethin about the stabilit of the sstem, we have to check whether the characteristic roots are reater or less than unit in absolute value. To determine whether λ are less than unit in absolute value, we use some additional conditions that are discussed below. CASE I: (TraceA) 4DetA 0 (implin that the roots are real) Consider the real line and take the two points + and as two reference points: ) There are si possibilities here: (i) Both λ lies in (-,), which implies that the equilibrium is stable. 7

6 (ii) One λ lies in (-,), the other one in (+, ), which implies that the equilibrium is a saddle point. (iii) One λ lies in (-,), the other one in (-, -), which implies that the equilibrium is a saddle point. (iv) One λ lies in (+, ), the other one in (-, -), which implies that the equilibrium is unstable. (v) Both λ lies in (+, ), which implies that the equilibrium is unstable. (vi) Both λ lies in (-, -), which implies that the equilibrium is unstable. But how do we know which of these cases are true? Can we determine that from the Trace and the Determinant of the co-efficient matri, as we did for the differential equation case? The answer is es, but in order to do that we need to know somethin more than just the sins of the Trace and the Determinant. To be more precise, we need to know whether λ lie on the same side or the opposite sides of and respectivel. Note that (i) If λ lie on the same side of + (either both lie to the left of +, or both to the riht), then ( λ )( λ) > 0, i.e., λ λ ( λ +λ) + > 0. (ii) If λ lie on the opposite sides of + (one to the left and one to the riht), then ( λ )( λ) < 0, i.e., λ λ ( λ +λ) + < 0. (iii) If λ lie on the same side of (either both lie to the left of, or both to the riht), then ( + λ )( + λ) > 0, i.e., λ λ + ( λ +λ) + > 0. (iv) if λ lie on the same side of (one to the left and the other to the riht), then ( + λ )( + λ) < 0, i.e., λ λ + ( λ +λ) + < 0. Notin that λ + λ = TraceA. λ = DetA, we can write these conditions in terms of TraceA and DetA as: (i) Det A TraceA + > 0 implies λ lie on the same side of +; (ii) Det A TraceA + < 0 implies λ lie on the opposite sides of +; (iii) Det A + TraceA + > 0 implies λ lie on the same side of ; (iv) Det A + TraceA + < 0 implies λ lie on the opposite sides of. Now let us consider all the possible cases: Case (I-a): Det A TraceA + > 0 ; Det A + TraceA + > 0 Then λ lie on the same side of + as well as. There are mutuall eclusive possibilities here: either λ, λ ( +, + ) ; or λ, λ (, ) ; or λ, λ (, + ). λ must lie in one of these sets; no other situation is possible. In order to know precisel in which set λ belon to, we look at the information iven about TraceA and DetA. DetA < λλ <, hence λ, λ (, + ) ; so the equilibrium is stable. DetA> and TraceA > 0 λ, λ ( +, + ) ; so the equilibrium is unstable. 8

7 DetA> and TraceA < 0 λ, λ (, ) ; so the equilibrium is unstable. Case (I-b): DetA TraceA+ < 0 ; Det A + TraceA + < 0 Then λ lie on the opposite sides of + as well as. The onl possibilit here : one λ lies in (+, ), the other one in (-, -), which implies that the equilibrium is unstable. Case (I-c): DetA TraceA+ > 0 ; DetA+ TraceA+ < 0 Then λ lie on the same side of +, but opposite sides of. The onl possibilit here is: one λ lies in (-,), the other one in (-, -), which implies that the equilibrium is a saddle point. Case (I-d): Det A TraceA + < 0 ; DetA+ TraceA+ > 0 Then λ lie on the opposite sides of +, but on the same side of. The onl possibilit here is: one λ lies in (-,), the other one in (+, +), which implies that the equilibrium is a saddle point. CASE II: (TraceA) 4DetA < 0 (implin that the roots are comple) If the roots are comple the have the eneral form u ± iv. Therefore, Trace A= λ+ λ = ( a+ ib) + ( a ib) = a Det A= λλ = ( a+ ib)( a ib) = a + b As was mentioned before, in the difference equation case with comple roots, stabilit depends on the term a + b. Thus we can determine the stabilit propert of the sstem b lookin at the determinant alone. If DetA= a + b <, then a + b <, so the sstem will show stable oscillations; If DetA= a + b >, then a + b >, so the sstem will show unstable oscillations; If DetA= a + b =, then a + b =, so the sstem will show uniform oscillations which are neither stable nor unstable. C. Phase Diaram Analsis: Sometimes alon with linearization technique, a diarammatic method is used in order to derive some qualitative conclusions about the behaviour of the dnamic sstem over time. This raphical method is known as the phase diaram technique and is often used to analse the behaviour of non-linear dnamic equations. (a) Phase portrait for sinle difference or differential equation: In the one dimensional (sinle equation) case, the phase diaram technique tpicall involves plottin the state variable in horizontal ais and the chanes in the value of the state variable alon the vertical ais. First consider a difference equation of the form t = f ( t ) The chane in the state variable is defined as 9

8 Δ = t t = ( ) t t f (8) Note that Δ > 0 means t > t, i.e., is increasin over time. Similarl, Δ < 0 means is decreasin over time. And Δ = 0 means is constant over time, which in turn implies that the sstem is at its stead state. Now we want to plot the chane in correspondin to different values of the state variable itself. From (8), we see that chane in is reflected in the difference between f ( ) and t. Thus we can plot these two functions separatel as f ) and t ˆ = t = ( t and observe their difference in order to et an idea about Δ. If we plot the o function ˆ = t, with t in the horizontal ais, we will et the 45 line. However plottin = f ( t ) with t in the horizontal ais requires certain information about the slope the curvature of the f function. The diaram below depicts a hpothetical f function as an eample. t- (45 o line) f( t- ) t- Note that in the diaram the f ( t ) curve intersects the 45 o line (representin t ) twice at and respectivel. At these two values Δ = f ( t ) t = 0; hence the represent the two stead state points. Also note that in a close neihbourhood of, if we take an value to the left of, f ( t ) lies below the 45 o line implin Δ < 0. Hence is decreasin in that reion. We draw an arrow pointin towards the left to indicate that is decreasin here. On the other hand, if we take an value to the riht of (in a small neihbourhood of ), f ( t ) lies above the 45 o line implin Δ > 0. Hence is increasin in that reion. Once aain we draw an arrow pointin towards the riht to indicate that is increasin here. Similarl, one can see that in a small neihbourhood of the second equilibrium point, Δ > 0 if we take an value lin to its left and Δ < 0 if we take an value lin to its riht. Hence we can draw the arrows accordinl. 30

9 Thus the diaram above complete with the direction of movements of (the arrows) in different reions enables us to derive raphicall that is a locall unstable equilibrium (because both the arrows on either side of point outwards indicatin that is movin awa from ) and is a locall stable equilibrium (because both the arrows in its either side point inwards indicatin that is movin towards ). The phase diaram for a differential equation can be constructed in an analoous manner. Consider the differential equation d = f (). The phase diaram plots the chane in correspondin to different values of the state variable. Here the chane in over time is directl measured b the f () function. If d d f ( ) > 0, > 0 - implin in increasin over time; if f ( ) < 0, < 0 - implin d in decreasin over time; and if f ( ) = 0, = 0 - implin remains constant (the stead state). If we are iven ceratin information reardin the slope and curvature of f (), we can draw the phase portrait for this differential equation. The diaram below aain depicts a hpothetical f function as an illustration: f() In the diaram the f () curve intersects the horizontal ais thrice, denotin three stead state points:, and respectivel. If we consider a small neihbourhood of the first equilibrium point, to its left f ( ) > 0 and to its riht f ( ) < 0 implin that in increasin to its left and decreasin to its riht. Thus aain we draw the arrows to indicate the direction of movement of. Similarl, we can find out the direction of movements of in close neihbourhoods of the other equilibrium points as well and draw the arrows accordinl. The complete phase diaram now tells us that and are locall stable equilibrium points while is locall unstable. (b) Phase portrait of a two dimensional sstem : In the two dimensional case, the phase diaram tpicall traces the curves alon which the values of the state variables remain unchaned over time. Usin these two curves as the reference points, one then identifies the reions where the state variables are chanin and determines the direction of chane for each of the state variable. 3

10 Let us first consider the sstem of difference equations: = f (, ) t t t t = ( t, t ) As before, we can denote the chanes in and respectivel as: Δ = f(, ) = fˆ (, ) t t t t t Δ = (, ) = ˆ (, ) t t t t t Now we trace the curves Δ = 0 and Δ = 0, measurin alon the horizontal ais and alon the vertical ais. d f Note that the slopes of these two curves in the (, plane is iven b = d Δ = 0 f d d = and respectivel. We cannot draw these curves unless we are iven Δ= 0 some information about these partial derivatives. For illustrative purposes, let us assume that f 0 ; f < ; < 0; < for all, > 0. > This implies that f ˆ = f > 0; f ˆ = f < 0; ˆ = < 0; ˆ = < 0. Then in the positive quadrant Δ = 0 is an upward slopin curve as shown below: Δ = 0 Δ > 0 Δ < 0 Let us now consider the reion above Δ = 0. Recall that alon an point on the Δ = 0 curve, f ˆ(, ) = 0. Now in the reion above this curve, for each value of, is hiher than the correspondin value alon Δ = 0. Since f ˆ > 0, it implies that in the reion above Δ = 0, f ˆ(, ) > 0. Thus Δ > 0 and is increasin in this reion. As before, we draw an arrow pointin to the riht to indicate that in increasin here. Usin analoous arument, we can show that in the reion below Δ = 0 curve, f ˆ(, ) < 0 implin Δ < 0 and is decreasin in this reion. Thus we draw an arrow here pointin to the left. On the other hand, iven the conditions on the partial derivatives, Δ = 0 is a downward slopin curve in the (, plane, as depicted below: 3

11 Δ < 0 Δ > 0 Δ = 0 Now consider the reion above Δ = 0 curve. For an, the value in this reion is hiher than the correspondin alon Δ = 0. Since ˆ < 0, it implies that in this reion < ˆ(, ) 0, i.e., Δ < 0. Thus in decreasin here. We draw an arrow pointin down to indicate the in decreasin in this reion. B analoous arument, we can show that in the reion below Δ = 0 curve, ˆ(, ) > 0; hence is increasin here and we draw an arrow point up to indicate that. Combinin these two diarams we can draw the complete phase portrait for this two dimensional sstem of difference equations as follows: Δ = 0 Δ = 0 The point of intersection between the two curves denote s the stead state such the at this point (and onl at this point) both Δ = 0 and Δ = 0. If we start from an other point, the direction of arrows tells us which wa and would move. In this specific case we find that all the arrows point towards the equilibrium; hence the equilibrium is stable here. If the directions of the arrows are such that the point outward, awa from the equilibrium, then the diaram tells us that the equilibrium is unstable. 33

12 If the arrows are such that from some reions the point towards the equilibrium, and from other reions, the point outward awa from the equilibrium, then we can conclude that the equilibrium is a saddle point. To draw the phase portrait of a two dimensional sstem differential equations, we can proceed in similar wa. Consider the followin sstem: d = f (, d = (, We first draw the curves f (, = 0 and (, = 0 in the (, plane and usin these two curves as reference points, determine the direction of movements of and for different reions lin above or below these two curves. Aain the point of intersection of the two curves would denote the stead state and the direction of arrows will tell us whether the stead state is stable, unstable or a saddle point, eactl as in difference equation case. 34