It is well known that there always exists an interaction of some kind between nonlinear oscillating systems. Minorsky stated that “Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interac- tion” [1]. In the monograph [1], we have investigated different interesting cases of interaction by using the effi- cient asymptotic method of nonlinear mechanics created by Krylov, Bogolyubov, and Mitropol’skii. The present paper introduces our study of the behavior of a van der Pol oscillator under parametric and forced excitations. The dynamical system under consideration is described by an ordinary nonlinear differential equation of the type (1.1). Section 1 is devoted to the case of small parameters. The amplitudes of nonlinear de- terministic oscillations and their stability are studied. Analytical calculations in combination with a computer are used to obtain amplitude curves, which show a very complicated form in Figs. 1.2–1.4. In Sec. 2, we study the chaotic phenomenon occurring in the system described by Eq. (1.1) without the assumption of the smallness of parameters. As is known, the fundamental characteristic of a chaotic system is its sensitivity to initial conditions. The diagnostic tool used in this work is the Lyapunov exponents. The positiveness of the largest Lyapunov exponent will help us to determine the values of parameters for which the chaotic motions are occurred. Chaotic attractors and associated power spectra will be presented.
Ukrainian Mathematical Journal, Vol 59, No 2, 2007 VAN DER POL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS Nguyen Van Dao,1 Nguyen Van Dinh,2 and Tran Kim Chi UDC 517.9 We study a van der Pol oscillator under parametric and forced excitations The case where a system contains a small parameter and is quasilinear and the general case (without the assumption of the smallness of nonlinear terms and perturbations) are studied In the first case, equations of the first approximation are obtained by the Krylov–Bogolyubov–Mitropol’skii technique, their averaging is performed, frequency-amplitude and resonance curves are studied, and the stability of the given system is considered In the second case, the possibility of chaotic behavior in a deterministic system of oscillator type is shown Introduction It is well known that there always exists an interaction of some kind between nonlinear oscillating systems Minorsky stated that “Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interaction” [1] In the monograph [1], we have investigated different interesting cases of interaction by using the efficient asymptotic method of nonlinear mechanics created by Krylov, Bogolyubov, and Mitropol’skii The present paper introduces our study of the behavior of a van der Pol oscillator under parametric and forced excitations The dynamical system under consideration is described by an ordinary nonlinear differential equation of the type (1.1) Section is devoted to the case of small parameters The amplitudes of nonlinear deterministic oscillations and their stability are studied Analytical calculations in combination with a computer are used to obtain amplitude curves, which show a very complicated form in Figs 1.2–1.4 In Sec 2, we study the chaotic phenomenon occurring in the system described by Eq (1.1) without the assumption of the smallness of parameters As is known, the fundamental characteristic of a chaotic system is its sensitivity to initial conditions The diagnostic tool used in this work is the Lyapunov exponents The positiveness of the largest Lyapunov exponent will help us to determine the values of parameters for which the chaotic motions are occurred Chaotic attractors and associated power spectra will be presented Case of Small Parameters In this section, we consider the case where the parameters are small The opposite case is investigated in the next section The smallness of parameters is characterized by introducing a small positive parameter ε In this case, the Krylov–Bogolyubov–Mitropol’skii asymptotic method [2, 3] is used for seeking approximate solutions and studying their stability Deceased Vietnam Academy of Sciences, Hanoi, Vietnam Published in Ukrains’kyi Matematychnyi Zhurnal, Vol 59, No 2, pp 206 – 216, February, 2007 Original article submitted August 14, 2006 0041–5995/07/5902–0215 © 2007 Springer Science+Business Media, Inc 215 NGUYEN VAN DAO, NGUYEN VAN DINH, 216 AND TRAN KIM CHI 1.1 Differential Equation of Oscillation and Its Stationary Solution The system under consideration is described by the equation { } x˙˙ + ω x = ε ∆ x − γ x + h(1 − k x ) x˙ + px cos 2ωt + e cos(ωt + σ ) , (1.1) where h > and k > are coefficients characterizing the self-excitation of a pure van der Pol oscillator, p > is the intensity of the parametric excitation, e > is the intensity of the forced excitation, and σ, ≤ σ ≤ π, is the phase shift between the parametric and forced excitations Below, two subcases are investigated separately for a weak parametric excitation when p < h2 and for a strong parametric excitation when p > h2 The solution of (1.1) is found in the form x = a cos ψ, x˙ = – a ω sin ψ, ψ = ω t + θ, (1.2) where a and θ are new variables that satisfy the following equations in the standard form: a˙ = – ε ∆ x − γ x + h(1 − k x ) x˙ + px cos 2ωt + e cos(ωt + σ ) sin ψ , ω { } ε ∆ x − γ x + h(1 − k x ) x˙ + px cos 2ωt + e cos(ωt + σ ) cos ψ θ˙ = – aω { (1.3) } Following the Krylov–Bogolyubov–Mitropol’skii method, in the first approximation these equations can be replaced by averaged ones: a˙ = – ε ε ka − a + pa sin 2θ + e sin (θ − σ ) , f0 = – hω 2ω 2ω { } ε ε a θ˙ = – g0 = – ∆a − γa3 + pa sin 2θ + e sin (θ − σ ) 2ω 2ω (1.4) The amplitude a and phase θ of stationary oscillations are determined from the equations a˙ = θ˙ = 0: ka f0 = hω − a + p a sin θ + e sin (θ – σ ) = 0, (1.5) g0 = ∆ a – γ a3 + p a cos θ + e cos ( θ – σ ) = These equations are equivalent to the following: ka f = f0 cos θ – g0 sin θ = p − ∆ + γ a a sin θ + hω − a cos θ – e sin σ = 0, (1.6) ka − a sin θ + p + ∆ − γ a a cos θ + e cos σ = 0, g = f0 sin θ + g0 cos θ = hω VAN DER P OL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS 217 or f = A sin θ + B cos θ – E = 0, (1.7) g = G sin θ + H cos θ – K = 0, where ka B = hω − 1 a, A = p − ∆ + γ a2 a , E = e sin σ, (1.8) ka G = hω − 1 a, H = p + ∆ − γ a a , K = – e cos σ 1.2 Amplitude-Frequency Relationship and Resonance Curve The characteristic determinants of Eqs (1.7) are as follows: A B G H = D = p − ∆ + γ a2 a ka − 1 a hω ka hω − 1 a p + ∆ − γ a2 a 2 2 2 2 ka − , = a p + ∆ − γa p − ∆ + γa − h ω 4 D1 = E B K H e sin σ = − e cos σ ka − 1 a hω p + ∆ − γ a2 a ka = ae p + ∆ − γ a sin σ + hω − cos σ , A D2 = G p − ∆ + γ a2 a = ka K hω − 1 a E (1.9) e sin σ − e cos σ ka = – ae p − ∆ + γ a cos σ + hω − sin σ Below, in the ( ∆, a )-plane we identify the regular region in which the characteristic determinant D is nonzero and the critical region in which D is identically zero NGUYEN VAN DAO, NGUYEN VAN DINH, 218 AND TRAN KIM CHI By solving Eqs (1.7) with respect to sin θ and cos θ and eliminating the phase θ, we obtain the amplitude–frequency relation W ( ∆, a ) = D12 D22 + ka − cos σ – D = a e p + ∆ − γ a sin σ + hω 2 ka + a e p − ∆ + γ a cos σ + hω − sin σ 2 2 2 2 2 2 ka – a p − ∆ + γa ∆ γ ω − = p+ − a − h (1.10) The regular part C1 of the resonance curve satisfies (1.10) and lies in the regular region, where D ≠ The critical part C2 of the resonance curve lies in the critical region, where ka or p – ∆ – γ a – h 2ω − 1 D = = 0, and satisfies the compatibility conditions D1 = or p + ∆ − γ a sin σ + hω ka − cos σ = 0, D2 = or p − ∆ + γ a cos σ + hω ka − sin σ = or 2 2 2 2 ka − ≥ e sin σ, a p − ∆ + γa + h ω or 2 2 2 2 2 ka − ≥ e cos σ a p + ∆ − γa + h ω and the trigonometric conditions 2 A + B ≥ E 2 (1.11) G2 + H2 ≥ K2 It is easy to see that the critical region is the resonance curve of a van der Pol oscillator under the action of a 2 parametric excitation without forced excitation ( e = ) For a weak parametric excitation ( p < h ), the resonance curve is an oval encircling the point A0 ∆ = γ a , a = a02 = , k VAN DER P OL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS 219 which is the representative point of the self-oscillation of the van der Pol oscillator This oval lies completely above the abscissa axis ∆ When the parametric excitation is strong enough ( p > h ), the critical oval enlarges and cuts the abscissa axis It follows from the compatibility conditions that ∆ = p cos σ + γa , ka hω − = – p sin σ (1.12) Hence, the compatibility point has the coordinates ∆ = ∆* = p cos σ + p sin 2σ a = a*2 = a02 − h + p cos 2σ γa , * The existence condition for the compatibility point is a*2 > or p sin 2σ < h + cos 2σ (1.13) Obviously, this condition is satisfied if sin σ < 0, i.e., π < σ < π 3π < σ < π or In the case where sin σ ≥ 0, i.e., ≤ σ ≤ π π ≤ σ ≤ or 3π (1.14) condition (1.13) can be transformed into Λ ( cos σ ) = p cos 2 σ + p h cos σ + h – p ≥ (1.15) The left-hand side of (1.15) is a trinomial in cos σ ∈ [ – 1, ] with the discriminant Γ = p ( h – h + p ) If h > – − p (the case h > + − p is not considered here), then Γ < and the trinomial Λ ( cos σ ) always has the same positive sign as its first coefficients, and condition (1.15) is satisfied by all values of σ in interval (1.14) If h ≤ – − p, then Γ ≥ and the trinomial Λ ( cos σ ) has either two simple roots or a double root The simple roots cos σ1, are cos σ1, = ( 2 − ph ± 2p ) Γ (1.16) NGUYEN VAN DAO, NGUYEN VAN DINH, 220 AND TRAN KIM CHI Fig 1.1 The heavy arcs give the values of σ for which the compatibility point I* exists It is noted that Λ ( ) = h ( + p ) > 0, Λ ( – ) = h ( – p ) ( p = ( ε ) ), and the numerical average of two roots S ph S = – satisfies the inequality – < < Hence, the two roots (1.16) lie in the interval [ – 1, ] Condi2 2p tion (1.15) yields cos σ ≤ cos σ2 or cos σ ≥ cos σ1 (1.17) Combining (1.17) with (1.14), we obtain ≤ σ ≤ σ1 or σ2 ≤ σ ≤ π , or π ≤ σ ≤ π + σ1 or π + σ2 ≤ σ ≤ 3π In summary, we have the following: if h > – − p2 , then the compatibility point I* exists for every σ ; if h ≤ – − p2 , then the compatibility point I* exists only for ≤ σ ≤ σ1, or σ2 ≤ σ ≤ π + σ1, or π + σ2 ≤ σ ≤ π (1.18) In Fig 1.1, the heavy arcs give the values σ for which the compatibility point I* exists when h ≤ – − p2 Since – − p2 is approximately equal to p 2, we can conclude the following: If p < h 2, i.e., if the parametric excitation is weak in comparison with self-excitation, then the critical oval D = lies completely above the abscissa axis ∆ The critical point I* always exists If p ≥ h 2, i.e., if the parametric excitation is strong enough, then the critical oval D = cuts the abscissa axis ∆ The critical point I* exists only for the values of σ lying in interval (1.18) Verifying the trigonometric conditions by substituting (1.12) into (1.11), we obtain VAN DER P OL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS p2 a*2 sin σ ≥ e sin σ, 221 p2 a*2 cos2 σ ≥ e cos σ (1.19) Since the right-hand sides of (1.19) are not equal to zero simultaneously, from (1.19) we find p2 a*2 ≥ e or a*2 ≥ e2 p2 (1.20) Hence, the compatibility point I* is only a critical point when the amplitude a is large enough, i.e., when the forced excitation is still not too strong in comparison with the parametric one 1.3 Forms of Resonance Curves To identify the forms of resonance curves we give in advance the values of h and k Then, for every chosen value of p, we change e and σ to have the resonance curves For example, for h = 0.1, k = 4, and ω = the self-excited oscillation of the van der Pol original system has the amplitude a02 = and is represented by the point A0 ∆ = γ, a02 = ( ) Case of Weak Parametric Excitation ( p < h ) As is known, in this case the critical oval D = 0, i.e., the resonance curve of the van der Pol oscillator under parametric excitation runs around the point A0 and lies entirely above the abscissa axis ∆ We take p = 0.05 and σ = For a weak forced excitation, i.e., when e is small enough, condition (1.20) is satisfied and the critical point I* with coordinates ∆ * = p + γ , a*2 = a02 = exists For sufficiently strong forced excitation, i.e., when e is large enough, the point I* is only a trivial compatibility point that does not belong to the resonance curve In Fig 1.2, the resonance curves 0, 1, 2, 3, 4, correspond to the linear case γ = for e = 0, 0.015, 0.0177, 0.050, 0.100, 0.120, respectively Curve is a critical oval Curve has two branches: branch C ′ ( ) lies near abscissa axis, and branch C ′′ lies higher and consists of two cycles, one of which C1′′ is outside and the other C2′′ is inside the critical oval These cycles are connected to one another at the critical node I* on the critical oval Increasing the forced excitation ( e ), the lower branch C ′ moves up The inner loop C2′′ of the upper branch is pressed while the outer loop C1′′ is expanded, but both loops are tied at the node I* For e ≈ 0.0177, the lower branch C ′ is connected with the outer loop C1′′ at the node J, and we have curve 2, where J is a singular point belonging to the regular region D ≠ For e larger than 0.0177, the singular point J disappears Then the lower branch and the outer loop are unified into one branch, which lies outside the critical oval We have the resonance curve As e increases further, the inner loop C2′′ continues to be pressed into I* and disappears when e = 0.1 (see the resonance curve 4) At this moment, I* is a returning point Curve corresponds to a very large value of forced excitation; the point I* is a trivial compatibility point that lies outside the resonance curve and does not belong to it Figures 1.3 show the resonance curves in the nonlinear case γ x , γ ≠ 0, for γ = – 0.1 (a) and γ = 0.1 (b) Curves 1, 3, and in these figures have the same values of parameters (except γ ) as for curves 1, 3, and in Fig 1.2 With a negative value of γ (see Fig 1.3a), resonance curves lean toward the left in comparison with the case γ = (Fig 1.2) Otherwise, resonance curves lean toward the right for a positive value of γ (see Fig 1.3b) This situation is common for nonlinear Duffing systems NGUYEN VAN DAO, NGUYEN VAN DINH, 222 AND TRAN KIM CHI Fig 1.2 Resonances curves for γ = 0, σ = 0, e = (curve 0), e = 0.015 (curve 1), e = 0.0177 (curve 2), e = 0.050 (curve 3), e = 0.100 (curve 4), e = 0.120 (curve 5) (a) (b) Fig 1.3 Resonance curves for γ = – 0.1 (a), = 0.1 (b), σ = 0, e = (curve 0), e = 0.015 (curve 1), e = 0.050 (curve 3), e = 0.12 (curve 5) Case of Strong Parametric Excitation ( p > h ) As before, we take h = 0.1 and k = 4, but p = 0.12 In this case, the critical oval is enlarged and cuts the abscissa axis ∆ In Fig 1.4a ( γ = ) and 1.4b ( γ = 0.1 ), the resonance curves correspond to σ = and e = 0.06 Curve has a cycle lying inside the critical oval and is connected with the outside branch by the critical point I* If only e increases, the inside cycle is tied and then disappears The critical point I * first becomes a returning point and then an isolated trivial compatibility point The resonance curve is the only outside branch that is moving up VAN DER P OL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS (a) 223 (b) Fig 1.4 Resonance curves for γ = (a), γ = 0.1 (b), σ = (curve 1), σ = π / 12 (curve 2), σ = π / (curve 3), σ = π / (curve 4) As σ changes, the critical point moves along the critical oval In Figs 1.4a and 1.4b, the resonance π π π , respectively We see curves 2, 3, and correspond to the values γ = (a), γ = 0.1 (b), and σ = , , 12 that, when σ increases, the critical point moves down, and the critical point I* becomes a returning one and then disappears Then the resonance curves separate into a cycle lying inside the oval and a branch lying outside this oval (curves and 4) 1.4 Stability Conditions To have a stability condition we use the variational equations by putting a = a0 + δ a and θ = θ0 + δ θ in (1.4) and neglecting the terms of higher degrees with respect to δ a and δ θ : d ε ∂f ε ∂f0 δθ , (δa) = – δa – dt 2ω ∂a 2ω ∂θ (1.21) ε ∂g ε ∂g0 d δθ , a0 (δθ) = – δa – 2ω ∂a 2ω ∂θ dt where a0 and θ0 are stationary values of the amplitude a and phase θ [the roots of Eqs (1.5)] The characteristic equation of system (1.21) is a0 ρ + ε ∂f0 ∂g0 ρ + ε ∂f ∂g − ∂f ∂g = + a 2ω ∂a ∂θ 4ω ∂a ∂θ ∂θ ∂a Hence, stability conditions for the stationary solutions a0 and θ0 have the form NGUYEN VAN DAO, NGUYEN VAN DINH, 224 AND TRAN KIM CHI ∂f ∂g S1 = a0 + ∂a ∂θ ka hk = a0 hω − + ω a02 + p sin 2θ – 4 { pa0 sin 2θ0 + e sin (θ0 − σ) } > (1.22) From the first equation of (1.5) we find sin θ0 ; then, by substituting it into (1.22), we get ( ) S1 = hω a0 ka02 − > or a02 > k This condition means that only oscillations with large amplitudes may be stable The second stability condition still has the following abbreviated form [1, 4]: S2 = ( a0 ∂W ∆, a0 D ∂a02 ( ) ( ) 2 D( ∆, a0 ) and W ( ∆, a0 ) have a determined sign ) > ( The curves D ∆, a02 = and W ∆, a02 = divide the plane P ∆, a02 functions ) into regions In each region, the Moving upwards along a straight line parallel to a02 and cutting the resonance curve at a point M, if we go from the region D W < (> 0) to the ordinate axis the region D W > (< 0), then the point M corresponds to the stable (unstable) oscillation Therefore, based on the sign distribution of the functions D and W in the P-plane, we can identify the stable and unstable branches of the resonance curve Case of Arbitrary Parameters Regular and Chaotic Solutions Let us go back to Eq (1.1), ignoring the assumption of the smallness of parameters, i.e., we consider the following differential equation: x˙˙ + ω x = ∆ x – γx + h(1 − k x ) x˙ + p x cos2 ω t + e cos ( ω t + σ ) (2.1) We fix parameters as follows: ω = 83, ∆ = 01, γ = 1, h = 1, k = 0.6, p = 0.001, and σ = ; we use e as a control parameter For different values of e, solutions of Eq (2.1) can be regular or chaotic To identify the regular or chaotic character of a solution, we can use various methods, such as the consideration of the sign of the largest Lyapunov exponent or building Poincaré sections [5 – 11] To construct a Poincaré section of an 2π orbit, we use the period T = of the external excitation force Then, the Poincaré section acts as a stroboω scope, freezing the components of motion commensurate with the period T If we have a collection of n discrete points on the Poincaré section, then the corresponding motion is periodic with period n T For example, for e = 5.09, the Poincaré section consists of three points (Fig 2.1a), and the motion is periodic with the period T; for e = 5.116, the Poincaré section consists of six points (Fig 2.1b), and the motion is periodic with period T If the Poincaré section does not consist of a finite number of discrete points, then the motion is aperiodic and may be chaotic (Fig 2.2) VAN DER P OL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS (a) 225 (b) Fig 2.1 Poincaré section: e = 5.09 (a), e = 5.116 (b) Fig.2.2 Poincaré section realized for e = 5.15 The periodic attractors and the corresponding power spectra realized for e = 5.09 and e = 5.116 are illustrated in Figs 2.3a and 2.3b The aperiodic attractor and its power spectrum realized for e = 5.15 are illustrated in Fig 2.4 In this case, the power spectrum has a continuous broadband character The Poincaré section has a distinctive form shown in Fig 2.2; it consists of about 8,000 points after the transition decays (the first 500 periods) To verify that the motion realized for e = 5.15 is chaotic, we need to show the sensitivity to initial conditions on this attractor We choose two points separated by d0 = 10 – and close to the attractor and examine evolutions initiated from them Figure 2.5 illustrates the variation of the separation d with time t The exponential growth of separation for 20 < t < 300 is clearly noticeable The separation saturates at the size of the attractor for t > 300 Therefore, there is a positive Lyapunov exponent associated with the chaotic orbit for e = 5.15 The evaluated largest Lyapunov exponent is λ ≈ 0.062 > (its calculation will be mentioned below) Evaluation of the Largest Lyapunov Exponent To evaluate the largest Lyapunov exponent in the case where ω = 0.83, ∆ = 0.01, γ = 1, h = 1, k = 0.6, p = 0.001, σ = 0, and e = 5.15, we represent Eq (2.1) in the form NGUYEN VAN DAO, NGUYEN VAN DINH, 226 AND TRAN KIM CHI Fig 2.3 Periodic attractors and associated power spectra: e = 5.09 (a), e = 5.116 (b) Fig 2.4 Chaotic attractor and associated power spectra for e = 5.15 x˙1 = x2 , ( ) x˙2 = – 0.6889x1 + 0.01 x1 – x13 + − 0.6 x 12 x + 0.002 x1 cos 2z + 5.15 cos z, (2.2) z˙ = 0.83 Let u = ( x1 , x2 , z ) be a three-dimensional vector and let u* = u* (t, u0 ) be a reference trajectory of system (2.2), where u0 is the initial condition The variational equation corresponding to this reference trajectory is ˙ = A η, η where η = u – u* and the matrix A depends on u* : A = − 0.6789 − x1* − 1.2 x1* x2* + 0.002 cos z* 1− 0.6 ( x1*)2 − 0.004 x1* sin z* − 5.15 sin z* VAN DER P OL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS 227 Fig 2.5 Sensitivity to initial conditions for e = 5.15 Fig 2.6 Time evolution of the largest Lyapunov exponent (one cycle = 2π / ω , ω = 0.83) This time evolution of the Lyapunov exponent is presented in Fig 2.6 The largest Lyapunov exponent is a positive number λ ≈ 0.062, which shows the chaotic character of the motion of system (2.2) This means that two trajectories starting closely to one another in the phase space will move exponentially away from each other for small times on the average: d ( t ) = d0 λ t , where d0 is the initial distance between two adjacent starting points at t = t0 and d is the distance between these two points at time t We return again to Fig 2.5, which shows how the distance d between evolutions initiated from two points separated by d0 = 10 – varies with time The separation grows exponentially within the range 20 < t < 300 before leveling off at the size of the attractor Conclusion The first section of the paper shows the efficiency of the asymptotic method created by Krylov, Bogolyubov, and Mitropol’skii in solving a complicated nonlinear problem Figure 1.2 presents different resonance curves in the “linear” case γ = There exists a special returning point I* on the resonance curves In the “nonlinear case” γ ≠ 0, the resonance curve leans toward the right for γ > (Fig 1.3b) and toward the left for γ < (Fig 1.3a), which is common for nonlinear Duffing systems 228 NGUYEN VAN DAO, NGUYEN VAN DINH, AND TRAN KIM CHI In the second section, it is seen that the chaotic phenomenon occurs in a deterministic system described by (2.1) The Poincaré section, chaotic attractor, and associated power spectra of the nonlinear oscillator (2.1) have been found The Fortran and Matlab software were used for calculating data and building the graphs This work was financially supported by the Council for Natural Science of Vietnam REFERENCES Nguyen Van Dao and Nguyen Van Dinh, Interaction between Nonlinear Oscillating Systems, Vietnam National University, Hanoi (1999) Yu A Mitropol’skii and Nguyen Van Dao, Applied Asymptotic Methods in Nonlinear Oscillations, Kluwer (1997) Yu A Mitropol’skii and Nguyen Van Dao, Lectures on Asymptotic Methods of Nonlinear Dynamics, Vietnam National University, Hanoi (2003) Nguyen Van Dao, Stability of Dynamical Systems, Vietnam National University, Hanoi (1998) A Nayfeh and B Balachandran, Applied Nonlinear Dynamics, Wiley (1995) Y Ueda, The Road to Chaos, Aerial Press (1992) T Kapitaniak and W H Steeb, “Transition to chaos in a generalized van der Pol’s equation,” J Sound Vibration, 143, No 1, 167–170 (1990) M Lakshmanan and S Rajasekar, Nonlinear Dynamics, Springer (2003) G L Baker and J P Gollub, Chaotic Dynamics An Introduction, Cambridge University Press (1990) 10 F C Moon, Chaotic Vibration, Wiley (1996) 11 K T Alligood, T D Sauer, and J A Yorke, Chaos An Introduction to Dynamical Systems, Springer, New York (1997) ... γ a , a = a02 = , k VAN DER P OL OSCILLATOR UNDER PARAMETRIC AND FORCED EXCITATIONS 219 which is the representative point of the self-oscillation of the van der Pol oscillator This oval lies... critical region is the resonance curve of a van der Pol oscillator under the action of a 2 parametric excitation without forced excitation ( e = ) For a weak parametric excitation ( p < h ), the resonance... is known, in this case the critical oval D = 0, i.e., the resonance curve of the van der Pol oscillator under parametric excitation runs around the point A0 and lies entirely above the abscissa