International Journal of Bifurcation and Chaos, Vol. 8, No. 3 (1998) 627–637
c World Scientific Publishing Company
SYMBOLIC COMPUTATION OF SECONDARY BIFURCATIONS IN A PARAMETRICALLY EXCITED SIMPLE PENDULUM ERIC A. BUTCHER and S. C. SINHA∗ Nonlinear Systems Research Laboratory, Department of Mechanical Engineering, Auburn University, Auburn, AL 36849, USA Received June 24, 1997; Revised October 30, 1997 A symbolic computational technique is used to study the secondary bifurcations of a parametrically excited simple pendulum as an explicit function of the periodic parameter. This is made possible by the recent development of an algorithm which approximates the fundamental solution matrix of linear time-periodic systems in terms of system parameters in symbolic form. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincar´e map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation boundaries in the parameter space. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it is shown how the bifurcation locations as well as the new linearized equations may be obtained in closed form as a function of the periodic parameter. Bifurcation diagrams are constructed and the results are compared with those obtained elsewhere using the point mapping method.
1. Introduction
determine the stability of time-invariant systems. Similar to the Routh–Hurwitz criteria for timeinvariant systems, various well-known criteria, such as Schur–Cohn, etc. may be used to extract the parameter-dependent stability and bifurcation relations of the time-periodic system from the FTM provided that this matrix can be symbolically obtained in terms of system parameters. Secondary bifurcations as a function of one or more parameters may thus be treated by repeating the linearization procedure and computing the new FTM after each bifurcation. It is well known that the exact solutions of periodic systems are possible only in a very limited number of cases due to the fact that, in general,
The study of dynamical systems governed by a set of nonlinear ordinary differential equations with periodic coefficients is of great theoretical and practical importance in various fields of science and engineering. A set of quasilinear equations generally arise when the system is expanded in a Taylor series about an equilibrium or a periodic motion of the system. The stability and bifurcation conditions are determined from the eigenvalues of the linearized system’s fundamental solution matrix evaluated at the end of the principal period (called the Floquet transition matrix (FTM)) in much the same way as the eigenvalues of a linear system matrix ∗
Corresponding author. E-mail:
[email protected] 627
628 E. A. Butcher & S. C. Sinha
such solutions do not exist. Two common asymptotic methods that have been used in the past to yield approximate stability boundaries in closed form for small systems include the perturbation method [Nayfeh, 1973] and the averaging technique [Sanders & Verhulst, 1985]. These methods are limited in application to systems with weak internal excitation since they are based on expanding the solution in terms of a small parameter that multiplies the time-periodic terms. When the excitation becomes strong, the parameter is no longer small (as is typically the case in secondary bifurcations) and the accuracy of the solution is very poor. Moreover, increasing the order of the approximation is usually very difficult and does not guarantee uniform convergence to the true solution. A recent approximation technique of Guttalu and Flashner [1996] for computing the FTM, or the linearized Poincar´e map, by truncated point mappings employs a variation of the standard Runge–Kutta numerical integration technique adapted to symbolic computation. Another technique to approximate the parameter-dependent FTM which has been developed recently involves the use of Picard iteration while expanding the time-periodic system matrix in shifted Chebyshev polynomials. This technique, which was introduced by Sinha and Butcher [1997], results in an algorithm for the computation of the fundamental solution matrix explicitly as a function of the system parameters and time and is easily implemented using a symbolic manipulator such as MATHEMATICA since it involves only simple matrix multiplications and additions. This is accomplished by employing the integration and product operational matrices associated with the polynomials and used in previous numerical computations by Sinha and Wu [1991] and Joseph et al. [1993]. By evaluating the fundamental solution matrix at the end of the principal period, the parameterdependent FTM is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space. Since these equations are expressed as homogeneous polynomials of the system parameters, they can be solved for one parameter once values of the others have been chosen. Similar to the truncated point mapping technique, this method can successfully be applied to periodic systems with strong internal excitations since it does not involve expansion
in terms of a small parameter. The primary bifurcations of the vertical equilibrium positions of a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force were analyzed via this procedure by Butcher and Sinha [1997]. In this study, the secondary bifurcations of the parametrically excited simple pendulum are similarly treated by repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution. For this purpose, it is shown how the bifurcation locations as well as the new linearized equations following each bifurcation may be obtained in closed form as a function of the periodic parameter. This is accomplished by using the information about the spectral density of the new solution and by assuming a parabolic-type bifurcation of the Poincar´e points. Bifurcation diagrams are then constructed and the results are found to agree with those obtained by Flashner and Hsu [1983] using the numerical point mapping method.
2. Symbolic Computation of the Floquet Transition Matrix Sinha and Butcher [1997] have outlined a technique for computing the fundamental solution matrix for a linear time-periodic dynamical system explicitly as a function of the system parameters and time via Picard iteration and expansion in shifted Chebyshev polynomials. When this matrix is evaluated at the end of the principal period, the parameterdependent Floquet transition matrix (FTM) is obtained, and hence it is possible to express the local stability and bifurcation conditions in a closed form. Two formulations were outlined: One applicable to general periodic systems and the other for systems in which the periodic linear system matrix contains constant terms. It was shown that the latter, called the “alternate formulation”, converges much faster than the former “general formulation”. The “general formulation” is based on expressing the linearized system x˙ = A(t, α)x,
x(0) = x0
(1)
in the equivalent integral form Z
t
0
x(t) = x +
A(τ, α)x(τ )dτ
(2)
0
where x(t) ∈ RN is the state vector, t ∈ R+ denotes time, and α ∈ RL is a parameter vector. The
Symbolic Computation of Secondary Bifurcations 629
method of Picard iteration is utilized to find the (k + 1)th approximation Z (k+1)
x
0
t
(t) = x + 0
Z
= I+ 0
Z
+ 0
t
A(τk , α)x(k) (τk )dτk Z
t
A(τk , α)dτk +
A(τk , α) · · ·
Z 0
τ1
0
A(τ, α) = T (τ )D(α)
"
Φ
(τ, α) = TˆT (τ ) Iˆ +
p X
(4)
! k−1
[L(α)]
τk 0
A(τk−1 , α)dτk−1 dτk + · · ·
where TˆT (τ ) is the N × N m Chebyshev polynomial matrix and D(α) is the N m × N Chebyshev coefficient matrix. This is inserted into Eq. (3) and the integration and product operational matrices [Sinha & Wu, 1991; Joseph et al., 1993] associated with the Chebyshev polynomials are employed to obtain an expression for the expansion of Φ(τ, α) in shifted Chebyshev polynomials as (p, m)
A(τk , α)
A(τ0 , α)dτ0 · · · dτk x0
where τ0 , . . . , τk are all dummy variables. The expression in brackets is an approximation to the fundamental solution matrix Φ(t, α) since it is truncated after a finite number of terms (iterations). After the period is normalized to unity via the transformation t = T τ , the normalized 1periodic system matrix A(τ, α) = A(τ + 1, α) is expanded in m shifted Chebyshev polynomials of the first kind valid in the interval [0, 1] as ˆT
Z
t
3. Method of Analysis The parametrically forced simple pendulum shown in Fig. 1 is now considered. It is assumed that the support of the pendulum is subjected to a sinusoidal motion in the vertical direction with frequency ω. The dimensionless equation of motion for the system is given by Flashner and Hsu [1983] as θ¨ + 2ξαθ˙ + (α2 − β sin 2tˆ) sin θ = 0
#
P (α) (5)
where (p, m) refers to the number of iterations and polynomials employed in the approximation, respectively. The N m × N Chebyshev coefficient ˆ L(α) = matrix B(α) is expressed in terms of I, T T ˆ ˆ ˆ G QD (α), and P (α) = G D(α) via simple matrix multiplications and additions of the operational matrices. The parameter-dependent FTM H(α) = Φ(τ = 1, α) is obtained by evaluating the fundamental solution matrix at the end of the principal period. For additional details on this algorithm (such as converence issues) as well as an outline of the “alternate formulation”, see [Sinha & Butcher, 1997]. For a detailed comparison with the truncated point mapping method of Guttalu and Flashner [1995, 1996] in terms of the required cpu time and memory, see [Butcher & Sinha, 1997].
(6)
where α = 2(ωn /ω), β = 4(A/L), ωn2 = g/L, c/M L2 = 2ξωn , and 2tˆ = ωt. Denoting x1 = θ, x2 = θ˙ and expanding sin θ in Taylor series about the vertical θ = 0 equilibrium position, the above equation may be written in state space
k=1
= TˆT (τ )B(α)
(3)
Fig. 1.
A parametrically excited simple pendulum.
630 E. A. Butcher & S. C. Sinha
form as ! x˙ 1 x˙ 2 "
=
0
1
−(α2 − β sin 2tˆ)
−2ξα
+
#
x1
!
x2
0
!
−(α2 − β sin 2tˆ)(−x31 /6 + x51 /120 − · · · ) (7)
Truncating the nonlinear terms, Eq. (7) reduces to the damped linear Mathieu equation where the system parameters are α, β, and ξ. For d = 2ξα = 0.31623, the H(α, β) matrix was computed using 32 Picard iterations and 20 Chebyshev polynomials and the corresponding bifurcation boundaries were found analytically from the relations det(I±H(α, β)) = 0. The positive sign corresponds to flip while the negative yields the fold bifurcation. After substituting values of α into these relations, 16th- degree polynomial equations in β result which may be solved for the boundaries once the complex roots and those outside the converged parameter region have been eliminated. The remaining roots lie on the stability boundaries which are shown in Fig. 2 within the vicinity of the primary 2:1 resonance. It is seen that the unstable flip bifurcation regions do not intersect the α-axis and have “lifted up” from their familiar position of intersecting position at a single point (at α2 = 1) in the undamped system. This feature of the damped system is consistent with the results shown by Newland [1989]. To compute specific bifurcation values of the periodic parameter, the case of α2 = 0.1 and
Fig. 2. Stability boundaries for the damped (ξ = 0.5, d = 0.31623) parametrically excited pendulum in the vicinity of the primary 2:1 resonance.
ξ = 0.5 (d = 2ξα = 0.31623) in Eq. (7) is considered. As the periodic parameter β is increased, the vertical equilibrium position remains stable until the point of primary flip bifurcation. The intersection of the α2 = 0.1 line and the flip stability boundary in Fig. 2 represents this point which may be found precisely by numerically solving the polynomial equation det(I + H(α, β)) = 0 for β with α2 = 0.1. The resulting bifurcation point was thus found to be at β = 1.7538 which agrees with the result found by Pandiyan [1994] and Pandiyan and Sinha [1995] using a trial and error approach until the desired accuracy was achieved. Thus far, the primary bifurcation boundaries of the vertical equilibrium position for the parametrically excited simple pendulum have been computed in the (α, β) plane and the β bifurcation value for α2 = 0.1 and ξ = 0.5 in particular. Now it will be shown how this technique may be used to calculate the exact parameter values at which the secondary bifurcations of Eq. (7) occur as the periodic parameter β is further increased. At the primary bifurcation point computed above, the vertical equilibrium position becomes unstable and a stable period 2T solution is born where T = π is the internal forcing period. This period 2T solution is plotted in the state space in Fig. 3 for four different values of β past the primary bifurcation point along with the Poincar´e points at tˆ = 0 and T . It is seen that the amplitude of the periodic solution grows with an increase in the periodic parameter, a result also found by Flashner and Hsu [1983]. The analytical expression for this 2T solution may be found by first assuming that the amplitude grows parabolically as a function of β. This
Fig. 3. The stable 2T -periodic solution in the phase plane of the parametrically excited pendulum for β = A) 1.76, B) 1.8, C) 2.0, and D) 2.2.
Symbolic Computation of Secondary Bifurcations 631
Fig. 4. Bifurcation diagrams for the parametrically excited pendulum showing the locations of the Poincar´e points both before and after the primary and secondary bifurcations. The solid lines represent stable solutions while the dashed lines represent unstable ones.
approximation is valid since the period doubling bifurcation produces a pitchfork as β changes. In fact, the reduced equation on the center manifold for this system at the point of flip bifurcation was found by Pandiyan and Sinha [1995] to be of the form u˙ = f (t)u3 + O(u5 )
(8)
where f (t) is a periodic function of time. This is of the same form as the well-known expression u˙ = µu − u3
(9)
in which the linear part is zero at the bifurcation value µ = 0. Since this expression results in a parabolic pitchfork bifurcation, the expressions p
x1 = θ = ±c1 β − 1.7538; p
x2 = θ˙ = ±c2 β − 1.7538
(10)
may be likewise assumed for the positions of the two Poincar´e points in the 2T solution where c1 and c2 are constants to be determined by using the (x1 , x2 ) values obtained via numerical integration for any given β > 1.7538. Following this procedure, for β = 2.2 (corresponding to Fig. 3D) the constants were found to be c1 = 0.78637 and c2 = 4.84917. The resulting bifurcation diagrams are shown in Fig. 4 where the solid lines (A1 , A2 ) represent the curves given by Eq. (10) and the points correspond to the actual positions of the Poincar´e points obtained via numerical integration for the four b values used in Fig. 3. It is seen that the expressions given by Eq. (10) are very close to the numerical values of the Poincar´e points shown. Finally, the Fourier transform (spectral density) of the numerically integrated periodic solution at β = 2.2 was computed where the magnitudes of the resulting Fourier coefficients are shown in Fig. 5.
632 E. A. Butcher & S. C. Sinha
Fig. 5.
Fourier transform (spectral density) of the numerically integrated solution for β = 2.2.
It is seen from this figure that the two frequency components are 2π/2T = 1 and 2π/(2T /3) = 3 and that the fundamental frequency is about ten times more dominant than is the third harmonic. After normalizing the Fourier coefficients using the positions of the Poincar´e points, the resulting expression for the 2T solution as a function of β was found to be x1 (tˆ, β) =
p
β − 1.7538(−9.9379 cos tˆ + 1.3586 sin tˆ
+ 0.1516 cos 3tˆ + 0.0710 sin 3tˆ)
(11)
This solution is plotted in Fig. 6 (dashed line) along
x˙ 1 x˙ 2
!
"
0 = −k1 (tˆ, β)
1 −2ξα
#
x1 x2
!
+
with the numerically integrated ones both in state space and as a time series for β = 2.2. It is seen that the solutions given by Eq. (11) are very close to the true solutions. To find the secondary bifurcation point as the periodic parameter β is further increased, Eq. (6) is expanded about the 2T -periodic solution given in Eq. (11). To make the expansion easier, however, Eq. (7) is actually used while keeping terms up to the fifth power in x since it was observed that the amplitude was greater than one radian at β = 2.2. The new equation for perturbations about x1 (tˆ, β) then takes the form
0 −k2 (tˆ, β)x21 − k3 (tˆ, β)x31 − k4 (tˆ, β)x41 − k5 (tˆ, β)x51
!
(12)
where k1 (tˆ, β) = (0.1−β sin 2tˆ) 1 1 × 1− x21 (tˆ, β)+ x41 (tˆ, β) ; 2 24 k2 (tˆ, β) = (0.1−β sin 2tˆ) 1 1 × − x1 (tˆ, β)+ x31 (tˆ, β) ; 2 12 1 1 2 ˆ ˆ ˆ k3 (t, β) = (0.1−β sin 2t ) − + x1 (t, β) ; 6 12 1 k4 (tˆ, β) = (0.1−β sin 2tˆ) x1 (tˆ, β) ; 24 1 k5 (tˆ, β) = (0.1 − β sin 2tˆ) 120
(13)
and it has been assumed that x1 (tˆ, β) satisfies Eq. (7). In fact, however, there is an error as shown in the time series for β = 2.2 of Fig. 7 in which it is seen that this “residual” is slightly less when using the quintic term in Eq. (7) than when it is neglected. The stability and bifurcation analysis of ˆ x1 (t, β) is performed by finding the Floquet transition matrix H(β) associated with the linear part of Eq. (12). First, the expression for k1 (tˆ, β) is expanded into a Fourier series which results in terms of the form cos 2ntˆ and sin 2ntˆ, n = 1, . . . , 7. The coefficients of these terms are simple monomials in β since the square root in Eq. (11) is factored out.
Symbolic Computation of Secondary Bifurcations 633
They are given by π 2 k1 (tˆ, β) = 2.59203 + 8.51813β − 7.07561β 2 + 0.96751β 3 + (−1.93382 − 26.60795β + 13.15646β 2 − 1.70127β 3 ) sin 2tˆ + (−0.65308 − 0.88396β + 1.15253β 2 − 0.24870β 3 ) cos 2tˆ + (0.27357 + 3.29268β − 2.65590β 2 + 0.39315β 3 ) sin 4tˆ + (−0.42594 − 9.55327β + 7.39969β 2 − 1.03434β 3 ) cos 4tˆ + (0.05043 + 2.09689β − 1.84867β 2 + 0.36301β 3 ) sin 6tˆ + (0.05409 + 1.34396β − 1.23911β 2 + 0.25956β 3 ) cos 6tˆ + (−0.00668 − 0.26516β + 0.28677β 2 − 0.07607β 3 ) sin 8tˆ + (0.00388 + 0.24854β − 0.26066β 2 + 0.06710β 3 ) cos 8tˆ + (−0.00017 − 0.01921β + 0.02207β 2 − 0.00631β 3 ) sin 10tˆ + (−0.00046 − 0.03293β + 0.03800β 2 − 0.01088β 3 ) cos 10tˆ + (0.00001 + 0.00229β − 0.00269β 2 + 0.00075β 3 ) sin 12tˆ + (−2.24 × 10−6 − 0.00083β + 0.00095β 2 − 0.00027β 3 ) cos 12tˆ + (0.00001β − 0.00001β 2 + 3.64 × 10−6 β 3 ) sin 14tˆ + (0.00006β − 0.00007β 2 + 0.00002β 3 ) cos 14tˆ
(14)
Fig. 7. The “residual” error of the analytical solution as a time series for β = 2.2. The solid line indicates results obtained by including up to the quintic term in Eq. (7) while the results for the dashed line only include the cubic term.
Fig. 6. Comparisons of the numerically integrated solution (solid lines) and the analytical one in Eq. (13) (dashed lines) in the phase plane and as a time series for β = 2.2.
These Fourier coefficients are graphed as a function of the periodic parameter both before and after the primary bifurcation point in Fig. 8. The corresponding time series of the linearized stiffness term for β = 2.2 is shown in Fig. 9 in which the horizontal dashed line signifies an average value of −2.612. Since the eigenvalues of H(β) = Φ(2T, β) are the squares of those for Φ(T, β), it is more
634 E. A. Butcher & S. C. Sinha
Fig. 8.
Fourier coefficients of the linearized stiffness both before and after the primary bifurcation.
efficient to symbolically evaluate the latter matrix using the linear part of Eq. (12) to find the secondary bifurcation point. Then the resulting multipliers may be squared to find the actual bifurcation route. By using the symbolic algorithm (with 10 iterations and 12 polynomials) for computing the parameter-dependent FTM for the new solution, it was found that Φ(T, β) has eigenvalues of −1 and
+1 at β = 1.75538 and 2.19042, respectively, so that the actual multipliers of Φ(2T, β) are +1 at both locations. The first value (1.75538) is very close to the previously computed value of the first (flip) bifurcation point, while the secondary bifurcation point analytically obtained as β = 2.19042 is very close to the true value (2.21) found numerically. It is suspected that the errors in the 2T
Symbolic Computation of Secondary Bifurcations 635
Fig. 9. The linearized stiffness as a time series for β = 2.2. The dashed line represents the average value of −2.612.
Fig. 11. Fourier transform (spectral density) of the numerically integrated (C1 , C2 ) solution for β = 2.26.
Fig. 10. Stable 2T -periodic solutions in the phase plane for β = 2.26 (after the secondary bifurcation).
Fig. 12. Comparisons of the numerically integrated solution (solid lines) and the analytical one in Eq. (17) (dashed lines) in the phase plane and as a time series for β = 2.26.
solution x ˆ1 (tˆ, β) shown in Fig. 7 are responsible for the slight discrepancy. The secondary bifurcation is thus a fold bifurcation to two new stable period 2T solutions (B1 , B2 ) and (C1 , C2 ) shown in Fig. 4, while the (A1 , A2 ) solution becomes unstable. This agrees with the results found by Flashner and Hsu [1983] and is also confirmed by the direct numerical
integration of the new stable 2T solutions shown in Fig. 10 at β = 2.26. The spectral density of the (C1 , C2 ) solution of Fig. 10 was computed and is shown in Fig. 11. It can be seen that the average value as well as the second harmonic are now nonzero. Following a similar procedure as described earlier, an approximate
636 E. A. Butcher & S. C. Sinha
expression for this new 2T solution was found as x ˜1 (tˆ, β) = −0.19687 − 0.40028 cos tˆ + 0.56202 sin tˆ + 0.00437 cos 2tˆ + 0.03350 sin 2tˆ + 0.06164 cos 3tˆ + 0.03447 sin 3tˆ +
p
β − 2.21(−0.24784 − 0.50391 cos tˆ + 0.70753 sin tˆ + 0.00550 cos 2tˆ
+ 0.04217 sin 2tˆ + 0.07760 cos 3tˆ + 0.04339 sin 3tˆ) This solution (dashed line) is plotted along with the numerically integrated one in the phase plane and as a time series in Fig. 12. However, an attempt to compute the linearized stiffness for this solution results in terms in which the square root appearing in Eq. (15) remains, in contrast to the previous x1 (tˆ, β) solution in which all terms in k1 (tˆ, β) were simple monomials in β. Since the symbolic algorithm for computing the FTM is less accurate in this situation (notice the significant discrepancy in the plots of x˜1 (tˆ, β) in Fig. 12), this approach is not pursued any further. The bifurcation diagram of Fig. 4, therefore, reflects the third bifurcation point at β ≈ 2.27 found numerically where the solutions (B1 , B2 ) and (C1 , C2 ) become unstable and the pendulum undergoes complete revolutions.
(15)
periodic parameter. Bifurcation diagrams were constructed and the results were found to agree with those obtained using the point mapping method. The convergence and accuracy of the solution depends on the number of iterations p and the number of Chebyshev polynomials m taken in the representation of the state vector and the elements of coefficients matrix. The details of convergence of this method for one and two degrees of freedom systems can be found in [Sinha & Butcher, 1997].
Acknowledgments Financial support for this work was provided by the Army Research Office, monitored by Dr. Gary L. Anderson under contract number DAAH04-94G0337.
4. Conclusions A symbolic computational study of the secondary bifurcations for a parametrically excited simple pendulum has been presented. This technique utilizes the recent symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet transition matrix (FTM), or the linear part of the Poincar´e map, was obtained. The subsequent use of wellknown criteria for the local stability and bifurcation conditions of equilibria and periodic solutions yielded the equations for the bifurcation surfaces in the parameter space as homogeneous polynomials of the periodic parameter. Since this method is not based on expansion in terms of a small parameter, it can successfully be applied to periodic systems whose internal excitation is strong, which is typically the case in secondary bifurcations. By repeating the linearization and computational procedure after each bifurcation of an equilibrium or periodic solution, it was shown how the bifurcation locations as well as the new linearized equations could be obtained in closed form as a function of the
References Butcher, E. A. & Sinha, S. C. [1998] “Symbolic computation of local stability and bifurcation surfaces for nonlinear time-periodic systems,” Nonlin. Dyn., to appear. Flashner, H. & Hsu, C. S. [1983] “A study of nonlinear periodic systems via the point mapping method,” Int. J. Numer. Methods Eng. 19, 185–215. Guttalu, R. S. & Flashner, H. [1996] “Stability analysis of periodic systems by truncated point mappings,” J. Sound and Vibration 189, 33–54. Joseph, P., Pandiyan, R. & Sinha, S. C. [1993] “Optimal control of mechanical systems subjected to periodic loading via Chebyshev polynomials,” Optimal Control Applications & Methods 14, 75–90. Nayfeh, A. H. [1973] Perturbation Methods (John Wiley, New York). Newland, D. E. [1989] Mechanical Vibration Analysis and Computation (John Wiley, New York). Pandiyan, R. [1994] Analysis and Control of Nonlinear Dynamic Systems with Periodically-Varying Parameters, Ph.D. dissertation, Auburn University, Auburn, AL. Pandiyan, R. & Sinha, S. C. [1995] “Analysis of timeperiodic nonlinear dynamical systems undergoing bifurcations,” Nonlin. Dyn. 8, 21–43.
Symbolic Computation of Secondary Bifurcations 637
Sanders, J. A. & Verhulst, F. [1985] Averaging Methods in Nonlinear Dynamical Systems (Springer-Verlag, New York). Sinha, S. C. & Wu, D.-H. [1991] “An efficient computational scheme for the analysis of periodic systems,” J. Sound and Vibration 151, 91–117.
Sinha, S. C. & Butcher, E. A. [1997] “Symbolic computation of fundamental solution matrices for timeperiodic dynamical systems,” J. Sound and Vibration 206(1), 61–85.
