robust method for experimental bifurcation analysis - Semantic Scholar
International Journal of Bifurcation and Chaos, Vol. 12, No. 8 (2002) 1909–1913 c World Scientific Publishing Company
ROBUST METHOD FOR EXPERIMENTAL BIFURCATION ANALYSIS GERRIT LANGER∗ and ULRICH PARLITZ† Drittes Physikalisches Institut, Universit¨ at G¨ ottingen, B¨ urgerstraße 42-44, D-37073 G¨ ottingen, Germany ∗ [email protected] † [email protected] Received September 16, 2001; Revised October 12, 2001 We present a robust method to locate and continue period-doubling, saddle-node and symmetrybreaking bifurcations of periodically driven experimental systems. The method is illustrated from results obtained for an electronic implementation of a Duffing oscillator. Keywords: Experimental; bifurcation; continuation.
1. Introduction One of the most important tasks in nonlinear dynamics is the determination of bifurcation sets of various dynamical systems. If the dynamical equations are known, bifurcation points in parameter space can be detected by solving some specific fixed point equations. Using suitable continuation algorithms these bifurcation points can be traced in parameter space in order to compute bifurcation curves or surfaces. A well written introduction into numerical bifurcation analysis and continuation can be found in [Seydel, 1988, 1991]. The fixed point conditions and their numerical solution are typically based on some derivatives of the flow with respect to state variables and parameters [Seydel, 1988, 1991; Parlitz, 1990]. These derivatives have to be approximated if the underlying equations are not known as it is the case for most experiments. Any experimental approach to bifurcation analysis requires a computer controlled experiment including manipulation of control parameters and online measurements of the current system dynamics. Using delay reconstruction one may, at least in principle, derive approximating (black-box) model equations which could then be used as starting
point for bifurcation analysis very similar to the theoretical case where the system’s equations are known. An example for this state space based approach was presented recently by Anderson et al. [1999] using simulated experiments. We found that state space based methods rely very much on the accuracy of the estimated derivatives and may be very susceptible with respect to noise. Therefore, we devised an algorithm for experimental bifurcation analysis that avoids modeling in (reconstructed) state space but is based directly on features of the measured signal such as autocorrelation functions, changes of magnitude, etc. In order to demonstrate our approach for experimental bifurcation analysis we have used an electronic implementation of Duffing’s equation x ¨ + δx˙ + x + x3 = γ cos(ωt) ,
(1)
where γ and ω are free control parameters and δ is fixed. Previous extensive numerical investigations revealed a rich bifurcation structure [Parlitz & Lauterborn, 1985; Parlitz, 1993] that turned out to be typical for periodically driven nonlinear oscillators [Scheffczyk et al., 1991; Parlitz et al., 1991; Eilenberger & Schmidt, 1992; Schmidt & Eilenberger, 1998]. Figure 1 shows the bifurcation
1909
1910 G. Langer & U. Parlitz true bifurcation curve
sn sn
pd
sb
.
log ω
structure of a resonance horn where saddle-node (sn), period-doubling (pd) and symmetry-breaking bifucations (sb) occur [Parlitz & Lauterborn, 1985; Parlitz, 1993].
2. Period-Doubling and Symmetry-Breaking Bifurcations 2.1. Detecting bifurcations Our algorithm for detecting period-doubling bifurcations is based on the assumption that the measured signal is periodic (i.e. transients decayed and we do not consider chaotic oscillations) and the period of the oscillations is an integer multiple of the driving period T = 2π/ω. The quantity that is monitored to detect bifurcations is the autocorrelation function. Let τ be the sampling time and {x(τ ), x(2τ ), . . . , x(Kτ )} a time series of length K and zero mean. Then the autocorrelation function reads as follows am = a(m · T ) N 1 X x(iτ )x(iτ + m · T ) , N i=1
(2)
where m is an integer. This function is evaluated only for small values of m. If, for instance, the
local detection procedure 1 2 3 4
detected
Fig. 1. The fundamental bifurcation pattern of Duffing’s oscillator (1) consisting of pd-, sb- and sn-bifurcations. The sn-bifurcation curves constitute the borders of the red regions where two coexisting attractors and hysteresis occur. In the orange area enclosed by the sb-bifurcation curve two asymmetrical orbits exist which are mirror images of each other. These orbits undergo a pd-bifurcation at the pd-curve when entering in the yellow region of period-doubled orbits. The bifurcation curves are computed numerically using a continuation algorithm.
continuation step
.
extrapolated
=
detection range and direction
. Parameter 2
log γ
pd
Start iterative steps to approximate the true curve
Parameter 1
Fig. 2.
Detection and continuation strategy.
signal has period-1 (in units of T ), then am is (almost) the same for different values of m. Also one would expect am to undergo a significant change when a period-doubling bifurcation occurs from period-n to period-2n with the consequence that now am 6= am+n but am = am+2n . The detection algorithm for pd-bifurcations evaluates the autocorrelation and propagates through parameter space with a predefined direction beginning at some set of parameters as illustrated in Fig. 2. When crossing a border between two areas of different periodicity the correlation coefficients am change. As soon as such an event is detected the algorithm inverts its propagation direction and halves its step size in order to increase accuracy for determining the position of the bifurcation point. If the accuracy is high enough the detection and refining procedure ceases and continuation begins (see Fig. 2). The direction of the first detection step is chosen to be towards higher periodicity. For the detection of a symmetry-breaking bifurcation a similar approach is used. To quantify symmetry from time series we monitor
bm = b m ·
T 2
N 1 X T m = (−1) x(iτ )x iτ + m · , N i=1 2
(3)
which for even m equals the autocorrelation am . For m = 1 it is the correlation of x(iτ ) with −x(iτ +(T /2)). For most excitation frequencies the ratio T /τ is not an integer. In these cases the required values x(iτ +(T /2)) are linearly interpolated.
Robust Method for Experimental Bifurcation Analysis 1911
A
900
B
pd
pd
γ /mV
Parameter 2
curved detection path with small enough radius
sb
100 2nd: reduced step, same directions 1st: full step, NO detection within range
2
3 =N max n steps with n-1
1
0.5 of full stepsize each
20 100
ω/2π /Hz
Parameter 1
Fig. 3.
Detection and continuation of turning points.
If we had perfect symmetry and no interpolation errors then b0 = b1 should hold. But since perfect symmetry is difficult to achieve experimentally we still call the oscillation symmetric if b1 /b0 > 0.95 (but not below 0.95).
1000
Fig. 4. Results of the continuation of pd-bifurcations and sb-bifurcations (the meaning of the colored regions is the same as in Fig. 1).
pd
pd
sn
sn pd
pd
2.2. Continuation Once a bifurcation point has been detected we start continuation in parameter space to obtain the desired bifurcation curve (comp. Fig. 1). Our continuation strategy is illustrated in Fig. 2. The algorithm moves forward along the bifurcation curve of interest and keeps a close distance to it in order to shorten the detection procedure. When at least two bifurcation points are already known, a third can be extrapolated linearly with a certain error depending on the true curve’s bending. The extrapolation step size is chosen adequately small to avoid leaving a local neighborhood where linear extrapolation generates errors within certain small limits. This neighborhood shrinks if bending of the curve increases which can be observed using at least three bifurcation points. When extrapolating a bifurcation point one also assumes a direction for the bifurcation curve to follow, which can be derived from the last two bifurcation points found or some larger set of succeeding points for higher order extrapolation. The direction of the steps of the local detection procedure are chosen to be orthogonal to the assumed direction of the curve to be continued. This shortens the detection process. In general the search direction changes during the continuation process and it turns, when the detected bifurcation points indicate a turn of the curve.
Fig. 5. Coexisting and disconnected branches emerging from a perturbed sb-bifurcation. The branches contain pdbifurcations that are shifted with respect to the corresponding pd-bifurcations of the other branch.
When approaching turning points continuation may become more complicated depending on how large the turn is compared to the continuation step size. Sharp turns of the order of magnitude of the step size require a different approach that is illustrated in Fig. 3. If the detection procedure was not successful and was terminated the algorithm repeats the continuation step with half of the previous step size as depicted in Fig. 3A. If for this new continuation step the detection of the bifurcation point again fails, the continuation step is reduced furthermore as shown in Fig. 3B. In order to cope with repeated failures of the detection the algorithm finally assumes a 1800 turn of the bifurcation curve and proceeds on a curved path around the conjectured turn (Fig. 3B). Along these curved paths detection is tried until the bifurcation curve is crossed again. Period-doubling and symmetry bifurcation curves obtained with this strategy can be seen in Fig. 4. Problems with the detection of symmetry bifurcations occurred due to the fact that the electronic
1912 G. Langer & U. Parlitz
1000
γ /mV
speaking sn-bifurcations of the slightly asymmetric electronic implementation. Nevertheless we decided to call them symmetry-breaking curves here to emphasize their correspondence to the (true) sb-curves obtained with the perfectly symmetrical numerical Duffing oscillator [Parlitz & Lauterborn, 1985; Parlitz, 1993]. Figure 6 shows both the pd-bifurcations already given in Fig. 4 (yellow) and the additional, coexisting pd-bifurcation regions (brown) shown qualitatively in the upper part of Fig. 5.
ω/2π /Hz
500
Fig. 6. Bifurcation curves of pd-bifurcations belonging to different branches (comp. Fig. 5). The brown areas belong to pd-bifurcations of coexisting attractors that are generated and destroyed by sn-bifurcations and are more difficult to reach and detect in experiments.
γ /mV
900
100
20 100 Fig. 7.
ω/2π /Hz
1000
Results of the continuation of sn-bifurcations.
implementation of Duffing’s equation was not perfectly symmetric. The deviation from a symmetric potential turned out to be strong enough to lead to an additional saddle node bifurcation and coexistence of different pd-bifurcation cascades. This observation was also confirmed by numerical simulations using a Duffing equation with a disturbed potential V (x) = x2 /2 + s(x3 /3) + x4 /4. For s = 0 the potential is symmetric and symmetric oscillations occur whose symmetry can be broken by a sb-bifurcation [Parlitz & Lauterborn, 1985]. For the asymmetric case s > 0 the sb-bifurcation turns into a sn-bifurcation and the locations of the pdbifurcations associated with the second branch are shifted as depicted in Fig. 5. In this sense the sb-bifurcation curves shown in Fig. 4 are strictly
3. Saddle-Node Bifurcations To detect and continue saddle-node bifurcations is quite a different task. Here we focused on sn-bifurcations that occur together with hysteresis loops. The measured sn-bifurcation curves are shown in Fig. 7. If one of two sn-bifurcations constituting a hysteresis loop occur, the rms-value of the signal x(t) changes rapidly and significantly. The second bifurcation of the hysteresis loop causes a change with opposite sign. Therefore, the algorithm tries to detect these sn-bifurcations as pairs and when it has detected one of the sn-bifurcations, it reverses its search direction and searches for the other where in general it uses a different but appropriately chosen starting point that lies in a close neighborhood of the assumed position of the bifurcation point in question. Each starting value has to be chosen carefully when operating in such a neighborhood of the curve to avoid crossing the curve by accident on the one hand and saving search time on the other hand. This is in contrast to the detection 900 pd
pd
γ /mV
100 100
100
sn
sb sn
20 100
ω/2π /Hz
1000
Fig. 8. Collection of all bifurcation curves measured for the electronic Duffing oscillator. (Colors are the same as in previous figures.)
Robust Method for Experimental Bifurcation Analysis 1913
can be used for bifurcation analysis of arbitrary driven, dissipative systems where period-doubling, symmetry-breaking or saddle-node bifurcations are to be detected and continued. Generalizations including Hopf bifurcations, transcritical bifurcations and other types of local bifurcations are possible. sn pd sb sn
Fig. 9. Numerically computed sn-, sb- and pd-bifurcation curves of the Duffing oscillator (1) for γ = 0.2. (Colors correspond to those used in Fig. 8.)
of pd-bifurcations, where the algorithm could operate in both step directions and the attractor is not lost if the search started too close to the curve and an accidental crossing happened. Continuation is accomplished by linearly extrapolating both bifurcation points simultaneously with a given step size that is not too large. The rest is the same as with the continuation of the other bifurcation types’ curves except for the treatment of turning points. Here two curves are continued. These two curves are expected to converge near the turning point forming a cusp.1 If the distance between both curves is smaller than some threshold the algorithm stops continuation and assumes that the cusp point is reached. As with the other continuation techniques the algorithm also terminates in cases of repeated detection failures. Figure 8 shows all measured bifurcation curves of the electronic Duffing oscillator. The overall structure is qualitatively similar to the bifurcation pattern of the numerical Duffing-oscillator [Parlitz & Lauterborn, 1985; Parlitz, 1993] that is shown for comparison in Fig. 9.
4. Conclusion Robust and simple to implement algorithms for experimental bifurcation analysis are presented and applied to an electronic oscillator. This approach
1
Acknowledgments G. Langer acknowledges support by the DFG (Graduiertenkolleg Str¨ omungsinstabilit¨ aten und Turbulenz). We thank all members of the nonlinear dynamics group at the Third Physical Institute for stimulating discussions and support.
References Anderson, J., Shvartsman, S., Fl¨ atgen, G., Kevrekidis, I., Rico-Mart´inez, R. & Krischer, K. [1999] “Adaptive method for the expoerimental detection of instabilities,” Phys. Rev. Lett. 82, 532–535. Eilenberger, G. & Schmidt, K. [1992] “Poincar´ e maps of Duffing-type oscillators and their reduction to circle maps: I. Analytic results,” J. Phys. A: Math. Gen. 25, 6335–6356. Parlitz, U. & Lauterborn, W. [1985] “Superstructure in ¨ + dx˙ + the bifurcation set of the Duffing equation x x + x3 = f cos(ωt),” Phys. Lett. A107(8), 351–355. Parlitz, U., Englisch, V., Scheffczyk, C. & Lauterborn, W. [1990] “Bifurcation structure of bubble oscillators,” J. Acoust. Soc. Am. 88(2), 1061–1077. Parlitz, U., Scheffczyk, C., Kurz, T. & Lauterborn, W. [1991] “On modeling driven oscillators by maps,” Int. J. Bifurcation and Chaos 1(1), 261–264. Parlitz, U. [1993] “Common dynamical features of periodically driven strictly dissipative oscillators,” Int. J. Bifurcation and Chaos 3, 703–715. Scheffczyk, C, Parlitz, U., Kurz, T., Knop, W. & Lauterborn, W. [1991] “Comparison of bifurcations structures of driven dissipative nonlinear oscillators,” Phys. Rev. A43(12), 6495–6502. Schmidt, K. & Eilenberger, G. [1998] “Poincar´ e maps of Dufiing-type oscillators and their reduction to circle maps: II. Methods and numerical results,” J. Phys. A: Math. Gen. 31, 3903–3927. Seydel, R. [1988] From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis (Elsevier Science, Amsterdam–NY). Seydel, R. [1991] “Tutorial on continuation, ” Int. J. Bifurcation and Chaos 1, 3–11.
The cusp point observed in parameter space is the result of the projection of a smooth curve in the full space spanned by parameters and state variables.
ROBUST METHOD FOR EXPERIMENTAL BIFURCATION ANALYSIS GERRIT LANGER∗ and ULRICH PARLITZ† Drittes Physikalisches Institut, Universit¨ at G¨ ottingen, B¨ urgerstraße 42-44, D-37073 G¨ ottingen, Germany ∗ [email protected] † [email protected] Received September 16, 2001; Revised October 12, 2001 We present a robust method to locate and continue period-doubling, saddle-node and symmetrybreaking bifurcations of periodically driven experimental systems. The method is illustrated from results obtained for an electronic implementation of a Duffing oscillator. Keywords: Experimental; bifurcation; continuation.
1. Introduction One of the most important tasks in nonlinear dynamics is the determination of bifurcation sets of various dynamical systems. If the dynamical equations are known, bifurcation points in parameter space can be detected by solving some specific fixed point equations. Using suitable continuation algorithms these bifurcation points can be traced in parameter space in order to compute bifurcation curves or surfaces. A well written introduction into numerical bifurcation analysis and continuation can be found in [Seydel, 1988, 1991]. The fixed point conditions and their numerical solution are typically based on some derivatives of the flow with respect to state variables and parameters [Seydel, 1988, 1991; Parlitz, 1990]. These derivatives have to be approximated if the underlying equations are not known as it is the case for most experiments. Any experimental approach to bifurcation analysis requires a computer controlled experiment including manipulation of control parameters and online measurements of the current system dynamics. Using delay reconstruction one may, at least in principle, derive approximating (black-box) model equations which could then be used as starting
point for bifurcation analysis very similar to the theoretical case where the system’s equations are known. An example for this state space based approach was presented recently by Anderson et al. [1999] using simulated experiments. We found that state space based methods rely very much on the accuracy of the estimated derivatives and may be very susceptible with respect to noise. Therefore, we devised an algorithm for experimental bifurcation analysis that avoids modeling in (reconstructed) state space but is based directly on features of the measured signal such as autocorrelation functions, changes of magnitude, etc. In order to demonstrate our approach for experimental bifurcation analysis we have used an electronic implementation of Duffing’s equation x ¨ + δx˙ + x + x3 = γ cos(ωt) ,
(1)
where γ and ω are free control parameters and δ is fixed. Previous extensive numerical investigations revealed a rich bifurcation structure [Parlitz & Lauterborn, 1985; Parlitz, 1993] that turned out to be typical for periodically driven nonlinear oscillators [Scheffczyk et al., 1991; Parlitz et al., 1991; Eilenberger & Schmidt, 1992; Schmidt & Eilenberger, 1998]. Figure 1 shows the bifurcation
1909
1910 G. Langer & U. Parlitz true bifurcation curve
sn sn
pd
sb
.
log ω
structure of a resonance horn where saddle-node (sn), period-doubling (pd) and symmetry-breaking bifucations (sb) occur [Parlitz & Lauterborn, 1985; Parlitz, 1993].
2. Period-Doubling and Symmetry-Breaking Bifurcations 2.1. Detecting bifurcations Our algorithm for detecting period-doubling bifurcations is based on the assumption that the measured signal is periodic (i.e. transients decayed and we do not consider chaotic oscillations) and the period of the oscillations is an integer multiple of the driving period T = 2π/ω. The quantity that is monitored to detect bifurcations is the autocorrelation function. Let τ be the sampling time and {x(τ ), x(2τ ), . . . , x(Kτ )} a time series of length K and zero mean. Then the autocorrelation function reads as follows am = a(m · T ) N 1 X x(iτ )x(iτ + m · T ) , N i=1
(2)
where m is an integer. This function is evaluated only for small values of m. If, for instance, the
local detection procedure 1 2 3 4
detected
Fig. 1. The fundamental bifurcation pattern of Duffing’s oscillator (1) consisting of pd-, sb- and sn-bifurcations. The sn-bifurcation curves constitute the borders of the red regions where two coexisting attractors and hysteresis occur. In the orange area enclosed by the sb-bifurcation curve two asymmetrical orbits exist which are mirror images of each other. These orbits undergo a pd-bifurcation at the pd-curve when entering in the yellow region of period-doubled orbits. The bifurcation curves are computed numerically using a continuation algorithm.
continuation step
.
extrapolated
=
detection range and direction
. Parameter 2
log γ
pd
Start iterative steps to approximate the true curve
Parameter 1
Fig. 2.
Detection and continuation strategy.
signal has period-1 (in units of T ), then am is (almost) the same for different values of m. Also one would expect am to undergo a significant change when a period-doubling bifurcation occurs from period-n to period-2n with the consequence that now am 6= am+n but am = am+2n . The detection algorithm for pd-bifurcations evaluates the autocorrelation and propagates through parameter space with a predefined direction beginning at some set of parameters as illustrated in Fig. 2. When crossing a border between two areas of different periodicity the correlation coefficients am change. As soon as such an event is detected the algorithm inverts its propagation direction and halves its step size in order to increase accuracy for determining the position of the bifurcation point. If the accuracy is high enough the detection and refining procedure ceases and continuation begins (see Fig. 2). The direction of the first detection step is chosen to be towards higher periodicity. For the detection of a symmetry-breaking bifurcation a similar approach is used. To quantify symmetry from time series we monitor
bm = b m ·
T 2
N 1 X T m = (−1) x(iτ )x iτ + m · , N i=1 2
(3)
which for even m equals the autocorrelation am . For m = 1 it is the correlation of x(iτ ) with −x(iτ +(T /2)). For most excitation frequencies the ratio T /τ is not an integer. In these cases the required values x(iτ +(T /2)) are linearly interpolated.
Robust Method for Experimental Bifurcation Analysis 1911
A
900
B
pd
pd
γ /mV
Parameter 2
curved detection path with small enough radius
sb
100 2nd: reduced step, same directions 1st: full step, NO detection within range
2
3 =N max n steps with n-1
1
0.5 of full stepsize each
20 100
ω/2π /Hz
Parameter 1
Fig. 3.
Detection and continuation of turning points.
If we had perfect symmetry and no interpolation errors then b0 = b1 should hold. But since perfect symmetry is difficult to achieve experimentally we still call the oscillation symmetric if b1 /b0 > 0.95 (but not below 0.95).
1000
Fig. 4. Results of the continuation of pd-bifurcations and sb-bifurcations (the meaning of the colored regions is the same as in Fig. 1).
pd
pd
sn
sn pd
pd
2.2. Continuation Once a bifurcation point has been detected we start continuation in parameter space to obtain the desired bifurcation curve (comp. Fig. 1). Our continuation strategy is illustrated in Fig. 2. The algorithm moves forward along the bifurcation curve of interest and keeps a close distance to it in order to shorten the detection procedure. When at least two bifurcation points are already known, a third can be extrapolated linearly with a certain error depending on the true curve’s bending. The extrapolation step size is chosen adequately small to avoid leaving a local neighborhood where linear extrapolation generates errors within certain small limits. This neighborhood shrinks if bending of the curve increases which can be observed using at least three bifurcation points. When extrapolating a bifurcation point one also assumes a direction for the bifurcation curve to follow, which can be derived from the last two bifurcation points found or some larger set of succeeding points for higher order extrapolation. The direction of the steps of the local detection procedure are chosen to be orthogonal to the assumed direction of the curve to be continued. This shortens the detection process. In general the search direction changes during the continuation process and it turns, when the detected bifurcation points indicate a turn of the curve.
Fig. 5. Coexisting and disconnected branches emerging from a perturbed sb-bifurcation. The branches contain pdbifurcations that are shifted with respect to the corresponding pd-bifurcations of the other branch.
When approaching turning points continuation may become more complicated depending on how large the turn is compared to the continuation step size. Sharp turns of the order of magnitude of the step size require a different approach that is illustrated in Fig. 3. If the detection procedure was not successful and was terminated the algorithm repeats the continuation step with half of the previous step size as depicted in Fig. 3A. If for this new continuation step the detection of the bifurcation point again fails, the continuation step is reduced furthermore as shown in Fig. 3B. In order to cope with repeated failures of the detection the algorithm finally assumes a 1800 turn of the bifurcation curve and proceeds on a curved path around the conjectured turn (Fig. 3B). Along these curved paths detection is tried until the bifurcation curve is crossed again. Period-doubling and symmetry bifurcation curves obtained with this strategy can be seen in Fig. 4. Problems with the detection of symmetry bifurcations occurred due to the fact that the electronic
1912 G. Langer & U. Parlitz
1000
γ /mV
speaking sn-bifurcations of the slightly asymmetric electronic implementation. Nevertheless we decided to call them symmetry-breaking curves here to emphasize their correspondence to the (true) sb-curves obtained with the perfectly symmetrical numerical Duffing oscillator [Parlitz & Lauterborn, 1985; Parlitz, 1993]. Figure 6 shows both the pd-bifurcations already given in Fig. 4 (yellow) and the additional, coexisting pd-bifurcation regions (brown) shown qualitatively in the upper part of Fig. 5.
ω/2π /Hz
500
Fig. 6. Bifurcation curves of pd-bifurcations belonging to different branches (comp. Fig. 5). The brown areas belong to pd-bifurcations of coexisting attractors that are generated and destroyed by sn-bifurcations and are more difficult to reach and detect in experiments.
γ /mV
900
100
20 100 Fig. 7.
ω/2π /Hz
1000
Results of the continuation of sn-bifurcations.
implementation of Duffing’s equation was not perfectly symmetric. The deviation from a symmetric potential turned out to be strong enough to lead to an additional saddle node bifurcation and coexistence of different pd-bifurcation cascades. This observation was also confirmed by numerical simulations using a Duffing equation with a disturbed potential V (x) = x2 /2 + s(x3 /3) + x4 /4. For s = 0 the potential is symmetric and symmetric oscillations occur whose symmetry can be broken by a sb-bifurcation [Parlitz & Lauterborn, 1985]. For the asymmetric case s > 0 the sb-bifurcation turns into a sn-bifurcation and the locations of the pdbifurcations associated with the second branch are shifted as depicted in Fig. 5. In this sense the sb-bifurcation curves shown in Fig. 4 are strictly
3. Saddle-Node Bifurcations To detect and continue saddle-node bifurcations is quite a different task. Here we focused on sn-bifurcations that occur together with hysteresis loops. The measured sn-bifurcation curves are shown in Fig. 7. If one of two sn-bifurcations constituting a hysteresis loop occur, the rms-value of the signal x(t) changes rapidly and significantly. The second bifurcation of the hysteresis loop causes a change with opposite sign. Therefore, the algorithm tries to detect these sn-bifurcations as pairs and when it has detected one of the sn-bifurcations, it reverses its search direction and searches for the other where in general it uses a different but appropriately chosen starting point that lies in a close neighborhood of the assumed position of the bifurcation point in question. Each starting value has to be chosen carefully when operating in such a neighborhood of the curve to avoid crossing the curve by accident on the one hand and saving search time on the other hand. This is in contrast to the detection 900 pd
pd
γ /mV
100 100
100
sn
sb sn
20 100
ω/2π /Hz
1000
Fig. 8. Collection of all bifurcation curves measured for the electronic Duffing oscillator. (Colors are the same as in previous figures.)
Robust Method for Experimental Bifurcation Analysis 1913
can be used for bifurcation analysis of arbitrary driven, dissipative systems where period-doubling, symmetry-breaking or saddle-node bifurcations are to be detected and continued. Generalizations including Hopf bifurcations, transcritical bifurcations and other types of local bifurcations are possible. sn pd sb sn
Fig. 9. Numerically computed sn-, sb- and pd-bifurcation curves of the Duffing oscillator (1) for γ = 0.2. (Colors correspond to those used in Fig. 8.)
of pd-bifurcations, where the algorithm could operate in both step directions and the attractor is not lost if the search started too close to the curve and an accidental crossing happened. Continuation is accomplished by linearly extrapolating both bifurcation points simultaneously with a given step size that is not too large. The rest is the same as with the continuation of the other bifurcation types’ curves except for the treatment of turning points. Here two curves are continued. These two curves are expected to converge near the turning point forming a cusp.1 If the distance between both curves is smaller than some threshold the algorithm stops continuation and assumes that the cusp point is reached. As with the other continuation techniques the algorithm also terminates in cases of repeated detection failures. Figure 8 shows all measured bifurcation curves of the electronic Duffing oscillator. The overall structure is qualitatively similar to the bifurcation pattern of the numerical Duffing-oscillator [Parlitz & Lauterborn, 1985; Parlitz, 1993] that is shown for comparison in Fig. 9.
4. Conclusion Robust and simple to implement algorithms for experimental bifurcation analysis are presented and applied to an electronic oscillator. This approach
1
Acknowledgments G. Langer acknowledges support by the DFG (Graduiertenkolleg Str¨ omungsinstabilit¨ aten und Turbulenz). We thank all members of the nonlinear dynamics group at the Third Physical Institute for stimulating discussions and support.
References Anderson, J., Shvartsman, S., Fl¨ atgen, G., Kevrekidis, I., Rico-Mart´inez, R. & Krischer, K. [1999] “Adaptive method for the expoerimental detection of instabilities,” Phys. Rev. Lett. 82, 532–535. Eilenberger, G. & Schmidt, K. [1992] “Poincar´ e maps of Duffing-type oscillators and their reduction to circle maps: I. Analytic results,” J. Phys. A: Math. Gen. 25, 6335–6356. Parlitz, U. & Lauterborn, W. [1985] “Superstructure in ¨ + dx˙ + the bifurcation set of the Duffing equation x x + x3 = f cos(ωt),” Phys. Lett. A107(8), 351–355. Parlitz, U., Englisch, V., Scheffczyk, C. & Lauterborn, W. [1990] “Bifurcation structure of bubble oscillators,” J. Acoust. Soc. Am. 88(2), 1061–1077. Parlitz, U., Scheffczyk, C., Kurz, T. & Lauterborn, W. [1991] “On modeling driven oscillators by maps,” Int. J. Bifurcation and Chaos 1(1), 261–264. Parlitz, U. [1993] “Common dynamical features of periodically driven strictly dissipative oscillators,” Int. J. Bifurcation and Chaos 3, 703–715. Scheffczyk, C, Parlitz, U., Kurz, T., Knop, W. & Lauterborn, W. [1991] “Comparison of bifurcations structures of driven dissipative nonlinear oscillators,” Phys. Rev. A43(12), 6495–6502. Schmidt, K. & Eilenberger, G. [1998] “Poincar´ e maps of Dufiing-type oscillators and their reduction to circle maps: II. Methods and numerical results,” J. Phys. A: Math. Gen. 31, 3903–3927. Seydel, R. [1988] From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis (Elsevier Science, Amsterdam–NY). Seydel, R. [1991] “Tutorial on continuation, ” Int. J. Bifurcation and Chaos 1, 3–11.
The cusp point observed in parameter space is the result of the projection of a smooth curve in the full space spanned by parameters and state variables.
