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Equivalent linear models for non-linear non-autonomous systems
Journal of Sound and Vibration (1975) 42(4): 441452
EQUIVALENT
LINEAR MODELS FOR NON-LINEAR
NON-AUTONOMOUS B. V.
SYSTEMS
DASARATHY
Computer Sciences Corporation,
Huntsville, Alabama 35802, U.S.A. (Received 15 November 1974, and in revisedform 5 April 1975) Equivalence between a class of non-linear non-autonomous systems of second order and a linear model of lower order is established through a differential transformation relation. It is shown that this equivalence can be established only under a certain constraint on the non-linear functional parameters of the given system. The equivalence automatically leads to the first integral which then can be analyzed further to obtain the response of the system. The feasibility of obtaining closed form solutions through such analysis is illustrated by considering certain sub-classes of systems, Further, the practical value of the technique is demonstrated through an example. 1. INTRODUCTION equivalent linear systems [l-5] has long been a popular analytical tool of research in the domain of non-linear analysis. Almost invariably, transformation techniques have been the basis for establishing such equivalence. The given non-linear and/or nonautonomous system in the (x-t) plane is mapped on to a new transformed (X-T) plane. The new dependent and independent variables are defined by transformation relations which could be either algebraic or differential expressions or both. While almost a limitless variety of transformation relations can be construed, the feasibility of these transformation rules permitting the solution in the new (X-T) plane to be mapped back onto the original (x-t) plane is to be ensured and this limits the form of the transformation relations to a few practical types, one of which was proposed in an earlier study [3] of a class of non-linear, autonomous systems of second order. This study was extended recently [4] to the domain of non-autonomous systems by modifying the transformation laws put forth in the earlier study. In all these studies, the equivalent linear model was of a second or higher order [S]. Here in this study, an equivalent linear model of first order is sought without restricting the equivalent model to be necessarily autonomous. Unlike in the previous studies, here the mapping back of the solutions in the transformed plane onto the original plane is a very simple and straightforward operation involving little or no algebraic calisthenics. This leads to the first integral which then is analyzed by using any one of the standard techniques of analysis of first order systems. The choice of the method is dictated, of course, by the form taken by the first integral which in turn is decided by the form of non-linear functional parameters of the given system of second order. However, for illustrative purposes, this exercise is gone through for a class of systems wherein the first integral can be deemed as an exact differential and the response of the system is obtained in closed form. Also, an example is appended to demonstrate the versatility of the technique in solving problems of practical interest.
Establishing
2. ANALYSIS Consider the class of second order, non-linear, equation of the type
non-autonomous
I + c(x, t) 3 + b(x, t) i + a(x, t) = 0, 447
systems governed
(.) e d( )/dt.
by an
(1)
448
B. V. DASARATHY
The objective of this study is to determine the sufficient conditions for establishing the equivalence of the given non-linear, non-autonomous system with a linear system of first order given by the equation X’ + N(T) X + M(T) = 0, (‘) e d( )/dT. (2) This equivalence is to be established through the transformation
2.1.
EQUIVALENCE
relations
X = P(X, t) f + 4(x, t),
(3)
T = r-(x, t).
(4)
THEOREM
The sufficient condition for the non-linear, non-autonomous system (equation (1)) and the linear system (equation (2)) to be equivalent under the transformation relations (equations (3) and (4)) is given by the semi-constraint relation? (aG)x = (bG - GA,
(5)
where G=G(x,r)=g(r)exp[Ic(x,f)dx]
(&)$00).
(6)
g(t) is an arbitrary twice differentiable function of f and subscripts denote corresponding partial derivatives. 2.2.
PROOF
Substitution of equations (3) and (4) in equation (2) leads to a second order, non-linear differential equation in x very similar to equation (1) in form and which will become identical to it under the following conditions of equivalence : Nr, + P&
= 4x, t),
(7)
Nr, + (pr + qx + (M + Nq) rJ/p = b(x, t),
(8)
(qr + (M + Nq) r,)/p = a(~, t).
(9)
Equation (7), on integration with respect to x, yields PNI = G(x, t),
(10)
where N,=exp(JN(r)dr).
(11)
Use of equations (10) and (11) in equations (8) and (9), to eliminate p and its derivatives, leads to the relations (12) W - G) = (M, + NI 4)x> aG = (M, + Nl q)t,
(13)
where MI =
I
MN, dr.
(14)
t The fact that the constraint in equation (5) involves an as yet unspecified function g(t) and, in addition, is an implicit differential relation (the explicit relation being obtained on integration with respect to one of the variables x or t giving rise to an additional function of the other variable t or x as the constant of integration), imparts a certain degree of laxity to the constraint relation. Hence equation (5) can be deemed as a semiconstraint. This is further elaborated later on in the text.
LINEAR MODELS FOR NON-LINEAR
SYSTEMS
449
Upon comparing equations (12) and (13), it can be observed that they are not at all independent and unlike equation (7) cannot be satisfied individually by judicious choice of the unspecified functions p, q, r, A4 and N. Thus it becomes clear that there exists a constraint, connecting the non-linear functions a, b and c of the non-linear system under consideration, which has to be satisfied before the equivalent linear system can be established. This constraint is to be derived by using equations (12) and (13). Partial differentiations of equation (12) with respect to t and equation (13) with respect to x, make the right-hand sides of these equations identical. This leads to the constraining equation (5) and hence the theorem. Alternatively. integration of equation (12) with respect to x (with t treated as constant) and integration of equation (13) with respect to t (with x treated as constant) also makes the right-hand sides of the equations identical. This leads to effectively the same constraint, but in a modified form, as MI + N1 q = / (bG - G,) dx + h(t) = J aG dt +f(x).
(15)
Partial differentiation of equation (15) successively with respect to x and I gives equation (51, the original form of the constraint. However, the laxity in the constraint, as mentioned earlier, is much more easily noticeable in equation (15), wherein there are two more unspecified arbitrary functions,f(x) and h(t), in addition to g(t), whose choice dictates the degree of constraint imposed on one of the non-linear functions a, b, or c. Now the solution of the equivalent linear differential equation (2) can be obtained directly as X=exp ( I--NdT
) PI
- /Mexp(jNdT)dT]
(16)
(K, is a constant of integration,) Equivalently, using equations (4), (I 1) and (14) gives N,X+M,=K,.
(17)
Substitution of equations (3) and (10) in equation (17) yields the first integral of equation (1) (subject to the constraint (5)) as Gi++(M,+Nlq)=K1,
(18)
where (MI + N,q) is given by equation (15). The fact that equation (18) is indeed the first integral of equation (1) easily can be verified by differentiating equation (18) with respect to t and using equations (12) and (13) to eliminate the two partial derivatives of (M, + N,q) arising from the above process of total differentiation. This leads back to equation (I) confirming thereby the validity of the analysis. Solution of equation (18) completes the study and gives the response of the given nonlinear system satisfying the constraint equation (5). Equation (18) being a first order differential equation, is far more tractable than equation (I). Depending on the actual forms of the functions CZ,b and c, which in turn dictates the form of equation (18), an appropriate choice, among the numerous methods of analysis available in the literature for solving first order equations, is to be made. While a detailed review of these methods to explore their suitability for solving equation (18) in its different possible forms is beyond the scope of this study, certain classes of problems wherein equation (18) assumes a simple form, and is readily solved, are discussed here. An obvious method that suggests itself for the solution of equation (IS) is that of exploring the feasibility of treating it as an exact differential of a function R(x, t), which when determined represents the response of the given system. One can rewrite equation (18) as Gdx+(M,+N,q-K,)dt=O.
(19)
450
B. V. DASARATHY
Equation (19) is an exact differential if and only if G, = (M, + Ni 4 - K,),.
(20)
Using equation (12) in equation (20) gives b = 2G,/G.
(21)
Equation (5) now can be rewritten, by using equation (21), as (22)
@G), = G,,, or, alternatively, upon integrating equation (22) with respect to x, a=
[J’
G. (23) II Equations (21) and (23) now represent the constraints on the given system, equation (l), and permit the determination of the closed form solution, R(x, t), as R(x,t)=
[
G,, dx + h,(t)
JGdx+Jhdr-(K,t+K,)
1
=O,
where Ki and K2 are the constants of integration to be chosen to satisfy the initial conditions of the system given by equation (1) under the constraints (21) and (23). An alternative approach to solution of equation (18) is that of exploring the feasibility of separation of variables. This can be achieved, while retaining K,, the constant of integration, as an arbitrary constant to be chosen to satisfy one of the initial conditions, if and only if (M, + iV,q) in equation (18) is a function of one variable x or t alone, and G is of the form g(W0). From equations (12), (13), and (6), the constraints on (M, + N,q) and G can be identified as constraints on the given non-linear functions u, b and c as c(x, t) = c(x), function of x only,
(24a)
and b(x, t) = gl(t), a function oft only,
a(x, t) = a(t)exp (-1 c(x)dx),
(24b)
or a(x, t) = 0, b(x, t) = (g + B(x))M).
(24c)
Equation (24a) along with equation (24b), or (24c), represent the constraints that have to be satisfied by the given non-linear system to be solved by this approach of separation of variables. In the sequel, the well known example of the Van der Pol oscillator is studied by a novel gambit of considering it as an approximation to a non-linear, non-autonomous system over a small range of the independent variable: namely the time, t. This permits obtaining an approximate first integral to the Van der Pol oscillator which can be integrated further by separation of variables. Thus, in the example to follow, the separation of variables approach is feasible, although the non-linear functions of the system considered do not fully satisfy the constraint relations (24). 2.3. EXAMPLE Consider the well known Van der Pol’s oscillator described by the equation Z--0(1 -x2)k+x=O.
(25)
LINEAR
MODELS FOR NON-LINEAR
SYSTEMS
451
Needless to mention, equation (25), in its present form, is not a particular case of the class of systems studied here as it does not satisfy the constraint relation (5). However, equation (25) can be viewed as a limiting case, with t -+ 0, of the equation -j _ e(e-“@ _ x’) z?+ e-“* x = 0.
(26)
Equation (26) in the range of 0 < t < ~9corresponds to equation (25) in an approximate sense. Thus a solution of equation (26) for small values of (t/f?) represents the response of the Van der Pol oscillator given by equation (25). Now, it is easily checked that equation (26), unlike equation (25), satisfies the semiconstraint relation (5) (with the unspecified functiong(t) being chosen as unity). This validates the applicability of the technique set forth in this study and hence, by using equations (I@, (12) and (13), the first integral of equation (26) can be derived as i + 8x3/3 - e e-r:e x = K,,
Kl = a constant of integration.
07)
Now, equation (27) has to be solved to obtain the final solution of equation (26). However, our interest in equation (26) is limited to small values of t (0 < t 4 0) and in this range of interest equation (27) reduces to i++x3/3-ox=K~,
o,<(tje)~l.
(28)
Equation (28) represents in an approximate sense, for small values of t/9 only, the first integral of the Van der Pol oscillator, and can be solved further by the separation ofvariables technique to give (29) J- dx/(K, + 0x - 0x3/3) = t + K,, K, = a second constant of integration. Equation (29) can be integrated for given values of 6 and initial conditions (which determine Kl and KJ to obtain the response of the Van der Pol’s oscillator in the range 0 < t Q 8. As t increases, equation (26) no longer corresponds to an oscillator and as t + a, it reduces to the degenerate case (this satisfies the constraint relation (24a)) I+Bx2~=0.
(30)
The first integral to equation (30) easily can be obtained either directly or from equation (27) (with t + m) as i+e8x3/3=Kl.
(31)
Equation (31) can be solved as before by separation of variables as dx/(K, i‘
- 8x3/3) = t + K,.
(32)
Thus, the example considered clearly demonstrates the wide range of applicability of the technique, which includes cases that do not, at first sight, appear to belong to the class of systems studied here. 3.
CONCLUDING REMARKS
The study presents the sufficient condition on the non-linear functional parameters of the system given by equation (1) for establishing a linear equivalent model of lower order. Establishing such equivalence leads to a straightforward determination of the first integral of the given non-linear second order differential equation with variable coefficients, by no means a trivial achievement. The feasibility of obtaining the second integral representing the response of the system in a closed form is illustrated only for a sub-class of systems, as an example, as the particular method to be adopted at this stage is dictated by the actual form resulting from the first integration. Thus no general approach can be spelled out and detailed analysis
452
B. V. DASARATHY
cannot be carried out unless and until the first integral is derived explicitly, which means the functions a, b and c have to be completely specified. As this study is aimed at establishing equivalent linear systems for wide classes of non-linear non-autonomous systems, at a single stroke, detailed derivation of the response of any one particular system is outside its domain of interest. However, the feasibility of such detailed analysis is demonstrated by tackling a relatively wide class of systems wherein the closed form solutions have been developed without having to prescribe the non-linear functions in explicit detail. Further, the versatile ability of this technique, in solving problems of practical interest, that are not directly recognizable as particular cases of the class of systems considered here, is demonstrated by an example. A significant point of this study is that the process of mapping the solution of the equivalent linear system in the new plane back on to the original plane involves no complexity whatsoever. This is because the equivalent system is one of a lower order and accordingly the transformation laws are also more explicit and direct, unlike in the case of the earlier studies in this field. The problem of solving the equivalent linear model itself is also very simple as the equivalent model is of first order. These attractive features, it is hoped, add to the value of this study in the analysis of non-linear, second order systems and will prove to be of interest to workers in this ever-green field of non-linear analysis. REFERENCES 1. J. M. THOMAS1952 Proceedings of the American Mathematical Society 3, 899-1903. Equations equivalent to a linear differential equation. 2. R. T. HERBST1956 Proceedings of the American Mathematical Society 7, 95-197. The equivalence of linear and non-linear equations. 3. B. V. DASARATHY1970 Journal of Sound and Vibration 11, 139-144. Analysis of a class of nonlinear systems. 4. B. V. DASARATHY 1974 International Journal of Systems Science 5, l-4. A differential transformation technique for the analysis of a class of non-linear time varying systems. 5. B. V. DASARATHY and P. SRINIVASAN 1969 SIAM Journal of Applied Mathematics 17, 511-515. On the synthesis of a class of second order non-linear differential equations.
EQUIVALENT
LINEAR MODELS FOR NON-LINEAR
NON-AUTONOMOUS B. V.
SYSTEMS
DASARATHY
Computer Sciences Corporation,
Huntsville, Alabama 35802, U.S.A. (Received 15 November 1974, and in revisedform 5 April 1975) Equivalence between a class of non-linear non-autonomous systems of second order and a linear model of lower order is established through a differential transformation relation. It is shown that this equivalence can be established only under a certain constraint on the non-linear functional parameters of the given system. The equivalence automatically leads to the first integral which then can be analyzed further to obtain the response of the system. The feasibility of obtaining closed form solutions through such analysis is illustrated by considering certain sub-classes of systems, Further, the practical value of the technique is demonstrated through an example. 1. INTRODUCTION equivalent linear systems [l-5] has long been a popular analytical tool of research in the domain of non-linear analysis. Almost invariably, transformation techniques have been the basis for establishing such equivalence. The given non-linear and/or nonautonomous system in the (x-t) plane is mapped on to a new transformed (X-T) plane. The new dependent and independent variables are defined by transformation relations which could be either algebraic or differential expressions or both. While almost a limitless variety of transformation relations can be construed, the feasibility of these transformation rules permitting the solution in the new (X-T) plane to be mapped back onto the original (x-t) plane is to be ensured and this limits the form of the transformation relations to a few practical types, one of which was proposed in an earlier study [3] of a class of non-linear, autonomous systems of second order. This study was extended recently [4] to the domain of non-autonomous systems by modifying the transformation laws put forth in the earlier study. In all these studies, the equivalent linear model was of a second or higher order [S]. Here in this study, an equivalent linear model of first order is sought without restricting the equivalent model to be necessarily autonomous. Unlike in the previous studies, here the mapping back of the solutions in the transformed plane onto the original plane is a very simple and straightforward operation involving little or no algebraic calisthenics. This leads to the first integral which then is analyzed by using any one of the standard techniques of analysis of first order systems. The choice of the method is dictated, of course, by the form taken by the first integral which in turn is decided by the form of non-linear functional parameters of the given system of second order. However, for illustrative purposes, this exercise is gone through for a class of systems wherein the first integral can be deemed as an exact differential and the response of the system is obtained in closed form. Also, an example is appended to demonstrate the versatility of the technique in solving problems of practical interest.
Establishing
2. ANALYSIS Consider the class of second order, non-linear, equation of the type
non-autonomous
I + c(x, t) 3 + b(x, t) i + a(x, t) = 0, 447
systems governed
(.) e d( )/dt.
by an
(1)
448
B. V. DASARATHY
The objective of this study is to determine the sufficient conditions for establishing the equivalence of the given non-linear, non-autonomous system with a linear system of first order given by the equation X’ + N(T) X + M(T) = 0, (‘) e d( )/dT. (2) This equivalence is to be established through the transformation
2.1.
EQUIVALENCE
relations
X = P(X, t) f + 4(x, t),
(3)
T = r-(x, t).
(4)
THEOREM
The sufficient condition for the non-linear, non-autonomous system (equation (1)) and the linear system (equation (2)) to be equivalent under the transformation relations (equations (3) and (4)) is given by the semi-constraint relation? (aG)x = (bG - GA,
(5)
where G=G(x,r)=g(r)exp[Ic(x,f)dx]
(&)$00).
(6)
g(t) is an arbitrary twice differentiable function of f and subscripts denote corresponding partial derivatives. 2.2.
PROOF
Substitution of equations (3) and (4) in equation (2) leads to a second order, non-linear differential equation in x very similar to equation (1) in form and which will become identical to it under the following conditions of equivalence : Nr, + P&
= 4x, t),
(7)
Nr, + (pr + qx + (M + Nq) rJ/p = b(x, t),
(8)
(qr + (M + Nq) r,)/p = a(~, t).
(9)
Equation (7), on integration with respect to x, yields PNI = G(x, t),
(10)
where N,=exp(JN(r)dr).
(11)
Use of equations (10) and (11) in equations (8) and (9), to eliminate p and its derivatives, leads to the relations (12) W - G) = (M, + NI 4)x> aG = (M, + Nl q)t,
(13)
where MI =
I
MN, dr.
(14)
t The fact that the constraint in equation (5) involves an as yet unspecified function g(t) and, in addition, is an implicit differential relation (the explicit relation being obtained on integration with respect to one of the variables x or t giving rise to an additional function of the other variable t or x as the constant of integration), imparts a certain degree of laxity to the constraint relation. Hence equation (5) can be deemed as a semiconstraint. This is further elaborated later on in the text.
LINEAR MODELS FOR NON-LINEAR
SYSTEMS
449
Upon comparing equations (12) and (13), it can be observed that they are not at all independent and unlike equation (7) cannot be satisfied individually by judicious choice of the unspecified functions p, q, r, A4 and N. Thus it becomes clear that there exists a constraint, connecting the non-linear functions a, b and c of the non-linear system under consideration, which has to be satisfied before the equivalent linear system can be established. This constraint is to be derived by using equations (12) and (13). Partial differentiations of equation (12) with respect to t and equation (13) with respect to x, make the right-hand sides of these equations identical. This leads to the constraining equation (5) and hence the theorem. Alternatively. integration of equation (12) with respect to x (with t treated as constant) and integration of equation (13) with respect to t (with x treated as constant) also makes the right-hand sides of the equations identical. This leads to effectively the same constraint, but in a modified form, as MI + N1 q = / (bG - G,) dx + h(t) = J aG dt +f(x).
(15)
Partial differentiation of equation (15) successively with respect to x and I gives equation (51, the original form of the constraint. However, the laxity in the constraint, as mentioned earlier, is much more easily noticeable in equation (15), wherein there are two more unspecified arbitrary functions,f(x) and h(t), in addition to g(t), whose choice dictates the degree of constraint imposed on one of the non-linear functions a, b, or c. Now the solution of the equivalent linear differential equation (2) can be obtained directly as X=exp ( I--NdT
) PI
- /Mexp(jNdT)dT]
(16)
(K, is a constant of integration,) Equivalently, using equations (4), (I 1) and (14) gives N,X+M,=K,.
(17)
Substitution of equations (3) and (10) in equation (17) yields the first integral of equation (1) (subject to the constraint (5)) as Gi++(M,+Nlq)=K1,
(18)
where (MI + N,q) is given by equation (15). The fact that equation (18) is indeed the first integral of equation (1) easily can be verified by differentiating equation (18) with respect to t and using equations (12) and (13) to eliminate the two partial derivatives of (M, + N,q) arising from the above process of total differentiation. This leads back to equation (I) confirming thereby the validity of the analysis. Solution of equation (18) completes the study and gives the response of the given nonlinear system satisfying the constraint equation (5). Equation (18) being a first order differential equation, is far more tractable than equation (I). Depending on the actual forms of the functions CZ,b and c, which in turn dictates the form of equation (18), an appropriate choice, among the numerous methods of analysis available in the literature for solving first order equations, is to be made. While a detailed review of these methods to explore their suitability for solving equation (18) in its different possible forms is beyond the scope of this study, certain classes of problems wherein equation (18) assumes a simple form, and is readily solved, are discussed here. An obvious method that suggests itself for the solution of equation (IS) is that of exploring the feasibility of treating it as an exact differential of a function R(x, t), which when determined represents the response of the given system. One can rewrite equation (18) as Gdx+(M,+N,q-K,)dt=O.
(19)
450
B. V. DASARATHY
Equation (19) is an exact differential if and only if G, = (M, + Ni 4 - K,),.
(20)
Using equation (12) in equation (20) gives b = 2G,/G.
(21)
Equation (5) now can be rewritten, by using equation (21), as (22)
@G), = G,,, or, alternatively, upon integrating equation (22) with respect to x, a=
[J’
G. (23) II Equations (21) and (23) now represent the constraints on the given system, equation (l), and permit the determination of the closed form solution, R(x, t), as R(x,t)=
[
G,, dx + h,(t)
JGdx+Jhdr-(K,t+K,)
1
=O,
where Ki and K2 are the constants of integration to be chosen to satisfy the initial conditions of the system given by equation (1) under the constraints (21) and (23). An alternative approach to solution of equation (18) is that of exploring the feasibility of separation of variables. This can be achieved, while retaining K,, the constant of integration, as an arbitrary constant to be chosen to satisfy one of the initial conditions, if and only if (M, + iV,q) in equation (18) is a function of one variable x or t alone, and G is of the form g(W0). From equations (12), (13), and (6), the constraints on (M, + N,q) and G can be identified as constraints on the given non-linear functions u, b and c as c(x, t) = c(x), function of x only,
(24a)
and b(x, t) = gl(t), a function oft only,
a(x, t) = a(t)exp (-1 c(x)dx),
(24b)
or a(x, t) = 0, b(x, t) = (g + B(x))M).
(24c)
Equation (24a) along with equation (24b), or (24c), represent the constraints that have to be satisfied by the given non-linear system to be solved by this approach of separation of variables. In the sequel, the well known example of the Van der Pol oscillator is studied by a novel gambit of considering it as an approximation to a non-linear, non-autonomous system over a small range of the independent variable: namely the time, t. This permits obtaining an approximate first integral to the Van der Pol oscillator which can be integrated further by separation of variables. Thus, in the example to follow, the separation of variables approach is feasible, although the non-linear functions of the system considered do not fully satisfy the constraint relations (24). 2.3. EXAMPLE Consider the well known Van der Pol’s oscillator described by the equation Z--0(1 -x2)k+x=O.
(25)
LINEAR
MODELS FOR NON-LINEAR
SYSTEMS
451
Needless to mention, equation (25), in its present form, is not a particular case of the class of systems studied here as it does not satisfy the constraint relation (5). However, equation (25) can be viewed as a limiting case, with t -+ 0, of the equation -j _ e(e-“@ _ x’) z?+ e-“* x = 0.
(26)
Equation (26) in the range of 0 < t < ~9corresponds to equation (25) in an approximate sense. Thus a solution of equation (26) for small values of (t/f?) represents the response of the Van der Pol oscillator given by equation (25). Now, it is easily checked that equation (26), unlike equation (25), satisfies the semiconstraint relation (5) (with the unspecified functiong(t) being chosen as unity). This validates the applicability of the technique set forth in this study and hence, by using equations (I@, (12) and (13), the first integral of equation (26) can be derived as i + 8x3/3 - e e-r:e x = K,,
Kl = a constant of integration.
07)
Now, equation (27) has to be solved to obtain the final solution of equation (26). However, our interest in equation (26) is limited to small values of t (0 < t 4 0) and in this range of interest equation (27) reduces to i++x3/3-ox=K~,
o,<(tje)~l.
(28)
Equation (28) represents in an approximate sense, for small values of t/9 only, the first integral of the Van der Pol oscillator, and can be solved further by the separation ofvariables technique to give (29) J- dx/(K, + 0x - 0x3/3) = t + K,, K, = a second constant of integration. Equation (29) can be integrated for given values of 6 and initial conditions (which determine Kl and KJ to obtain the response of the Van der Pol’s oscillator in the range 0 < t Q 8. As t increases, equation (26) no longer corresponds to an oscillator and as t + a, it reduces to the degenerate case (this satisfies the constraint relation (24a)) I+Bx2~=0.
(30)
The first integral to equation (30) easily can be obtained either directly or from equation (27) (with t + m) as i+e8x3/3=Kl.
(31)
Equation (31) can be solved as before by separation of variables as dx/(K, i‘
- 8x3/3) = t + K,.
(32)
Thus, the example considered clearly demonstrates the wide range of applicability of the technique, which includes cases that do not, at first sight, appear to belong to the class of systems studied here. 3.
CONCLUDING REMARKS
The study presents the sufficient condition on the non-linear functional parameters of the system given by equation (1) for establishing a linear equivalent model of lower order. Establishing such equivalence leads to a straightforward determination of the first integral of the given non-linear second order differential equation with variable coefficients, by no means a trivial achievement. The feasibility of obtaining the second integral representing the response of the system in a closed form is illustrated only for a sub-class of systems, as an example, as the particular method to be adopted at this stage is dictated by the actual form resulting from the first integration. Thus no general approach can be spelled out and detailed analysis
452
B. V. DASARATHY
cannot be carried out unless and until the first integral is derived explicitly, which means the functions a, b and c have to be completely specified. As this study is aimed at establishing equivalent linear systems for wide classes of non-linear non-autonomous systems, at a single stroke, detailed derivation of the response of any one particular system is outside its domain of interest. However, the feasibility of such detailed analysis is demonstrated by tackling a relatively wide class of systems wherein the closed form solutions have been developed without having to prescribe the non-linear functions in explicit detail. Further, the versatile ability of this technique, in solving problems of practical interest, that are not directly recognizable as particular cases of the class of systems considered here, is demonstrated by an example. A significant point of this study is that the process of mapping the solution of the equivalent linear system in the new plane back on to the original plane involves no complexity whatsoever. This is because the equivalent system is one of a lower order and accordingly the transformation laws are also more explicit and direct, unlike in the case of the earlier studies in this field. The problem of solving the equivalent linear model itself is also very simple as the equivalent model is of first order. These attractive features, it is hoped, add to the value of this study in the analysis of non-linear, second order systems and will prove to be of interest to workers in this ever-green field of non-linear analysis. REFERENCES 1. J. M. THOMAS1952 Proceedings of the American Mathematical Society 3, 899-1903. Equations equivalent to a linear differential equation. 2. R. T. HERBST1956 Proceedings of the American Mathematical Society 7, 95-197. The equivalence of linear and non-linear equations. 3. B. V. DASARATHY1970 Journal of Sound and Vibration 11, 139-144. Analysis of a class of nonlinear systems. 4. B. V. DASARATHY 1974 International Journal of Systems Science 5, l-4. A differential transformation technique for the analysis of a class of non-linear time varying systems. 5. B. V. DASARATHY and P. SRINIVASAN 1969 SIAM Journal of Applied Mathematics 17, 511-515. On the synthesis of a class of second order non-linear differential equations.