Identification of minimal parameters for optimal suppression of chaos in dissipative driven systems

Taming chaos arising from dissipative non-autonomous nonlinear systems by applying additional harmonic excitations is a reliable and widely used procedure nowadays. But the suppressory effectiveness of generic non-harmonic periodic excitations continues to be a significant challenge both to our theoretical understanding and in practical applications. Here we show how the effectiveness of generic suppressory excitations is optimally enhanced when the impulse transmitted by them (time integral over two consecutive zeros) is judiciously controlled in a not obvious way. Specifically, the effective amplitude of the suppressory excitation is minimal when the impulse transmitted is maximum. Also, by lowering the impulse transmitted one obtains larger regularization areas in the initial phase difference-amplitude control plane, the price to be paid being the requirement of larger amplitudes. These two remarkable features, which constitute our definition of optimum control, are demonstrated experimentally by means of an analog version of a paradigmatic model, and confirmed numerically by simulations of such a damped driven system including the presence of noise. Our theoretical analysis shows that the controlling effect of varying the impulse is due to a subsequent variation of the energy transmitted by the suppressory excitation.

1. Figure S2: Comparison between the elliptic SE and its two-and three-harmonics approximations over a period for four values of the shape parameter. Plots of the elliptic SE (Eq. S1, dashed line), its two-harmonics approximation S 2 (t) ≡ a 0 (m) sin (ωt + ϕ) + a 1 (m) sin (3ωt + 3ϕ) (cf. Eqs. S4 and S5, solid line), and its three-harmonics approximation S 3 (t) ≡ a 0 (m) sin (ωt + ϕ) + a 1 (m) sin (3ωt + 3ϕ) + a 2 (m) sin (5ωt + 5ϕ) (cf. Eqs. S4 and S5, thin solid line ) versus time for four values of the shape parameter: Melnikov analysis (MA) [2,3] by solely retaining the first harmonic of the Fourier expansion (Eq. S4): S2): Third, regarding numerical simulations, we considered the entire Fourier expansion of the elliptic SE in order to obtain useful information concerning the effectiveness of the approximations used in the theoretical analysis and experiments (cf. Eqs. S6 and S7, respectively).

B. Chaotic threshold from Melnikov analysis
Melnikov introduced a function (now known as the Melnikov function (MF), M (t 0 )) which measures the distance between the perturbed stable and unstable manifolds in the Poincaré section at t 0 . If the MF presents a simple zero, the manifolds intersect transversally and chaotic instabilities result. See Refs. [2,3] for more details about MA. Regarding Eq. (2) in the main text, note that keeping with the assumption of the MA, it is assumed that one can write δ = εδ, γ = εγ, η = εη where 0 < ε ≪ 1 while δ, γ, η, β, ω are of order one. Thus, the application of MA to Eq. (2) in the main text yields the MF where the coefficients a p (m) are given by Eq. S5, and where the positive (negative) sign refers to the right (left) homoclinic orbit of the underlying conservative Duffing oscillator (δ = η = γ = 0): .
x 0,± (t) = ∓ 2 β sech (t) tanh (t) . (S14) Let us assume that, in the absence of any SE (η = 0), the damped driven two-well Duffing oscillator (Eq. 2 in the main text) presents chaotic behaviour for which the respective MF, has simple zeros, i.e., D A or where the equal sign corresponds to the case of tangency between the stable and unstable manifolds [3]. If we now let the SE act on the Duffing oscillator such that B * A − D, with then this relationship represents a sufficient condition for M ± (t 0 ) to change sign at some t 0 . Thus, a necessary condition for M ± (t 0 ) to always have the same sign is Since a p (m) > 0, b p (T ) > 0, p = 0, 1, 2, ..., one straightforwardly obtains and hence, .

(S21)
Note that Eq. S20 provides a lower threshold for the amplitude of the SE. Similarly, an upper threshold is obtained by imposing the condition that the SE may not enhance the initial chaotic state (i.e., it does not increase the (initial) gap from the homoclinic tangency condition), and hence, which is a necessary condition for M ± (t 0 ) to always have the same sign. Thus, the suitable (suppressory) amplitudes of the SE must satisfy while the width of the range of suitable amplitudes reads  This means that ever lower amplitudes η min can suppress chaos as the impulse transmitted by the SE approaches its maximum value, whereas the corresponding suppressory ranges ∆η also decrease in the same way as η min owing to the impulse-induced enhancement of the chaos-inducing effectiveness of the SE. This dependence of η max , η min , ∆η on the SE's impulse represents a genuine feature of the impulse-induced chaos-control scenario.
Regarding suitable values of the initial phase difference ϕ, note that ϕ determines the relative phase between M ± 0 (t 0 ) and irrespective of the shape parameter value. We, therefore, conclude from previous theory [4] that a sufficient condition for η min < η < η max to also be a sufficient condition for suppressing chaos is that M ± 0 (t 0 ) and are in opposition. This yields the optimum suppressory values for all m ∈ [0, 1] in the sense that they allow the widest amplitude ranges for the elliptic SE.
To obtain an useful analytical estimate of the boundaries of the regions in the ϕ − η control plane where chaos is suppressed, we assume the first-harmonic approximation given by Eq. S6 instead of the entire Fourier expansion (cf. Eq. S4) in the remainder of this section. Indeed, recall that the value m impulse max ≃ 0.717 at which the SE's impulse presents a single maximum is very close to the value m = m max (n = 0) ≃ 0.642 where the amplitude a 0 (m) (cf. Eq. S5) presents a single maximum (see Fig. S1). Thus, we apply MA to the effective MF while the effectiveness of the first-harmonic approximation (η > 0) at suppressing chaos will be examined by considering for example the effective MF M + ef f (t 0 ) (the analysis of M − ef f (t 0 ) is similar and leads to the same conclusions). To this end, it is convenient to use the normalized MF where R ′ ≡ A/D, R ′′ ≡ B 0 /D. If one now lets the first-harmonic approximation act on the system such that this relationship represents a sufficient condition for M + n (t 0 ) to be negative (or null) for all t 0 . The equals sign in Eq. S30 yields the boundary of the region in the ϕ − η plane where chaos is suppressed: with γ > γ th (cf. Eq. S16), and where the two signs before the square root correspond to the two branches of the boundary (see Fig. S5). The following remarks may now be in order.
First, the boundary function (Eq. S31) represents two loops encircling the regularization regions in the ϕ − η plane which are symmetric with respect to the optimal suppressory values respectively, i.e., those values of the initial phase difference for which the range of suitable suppressory values of η is maximum. As expected, they are the same suppressory values than those found in the exact case of representing the elliptic SE by its entire Fourier expansion (cf. Eq. S26).
Second, the area, A R , enclosed by the boundary function (Eq. S31) is straightforwardly obtained from previous theory [4]: .
Observe that one finds A R → 0 as δ → 0, which corresponds to the limiting Hamiltonian case, as expected. More importantly, the normalized regularization area presents, as a function of the shape parameter, a single minimum at the m value where a 0 (m) presents a single maximum (see Fig. S1): m max (n = 0) ≃ 0.642, which is very close to m impulse max ≃ 0.717. This inverse dependence of the regularization area on the SE's impulse represents a genuine feature of the impulse-induced chaos-control scenario.
Third, the regularization area shrinks as the ratio γ th /γ diminishes, which means that the impulse-induced chaos-control scenario is sensitive to the strength of the initial chaotic state in the sense of its proximity to the threshold condition (cf. Eq. S16). text has the associated energy equation where, for the sake of convenience, we introduced the shift t → t+T /4, and hence ϕ → ϕ−π/2, and where E(t) ≡ (1/2) nT .
Now, if we consider fixing the parameters (δ, γ, β, T ) for the Duffing oscillator to undergo chaotic behaviour at η = 0, there always exists an n = n * such that the energy increment ∆E ≡ E (n * T + T /2) − E(n * T ) is positive before chaotic escape from one of the two potential wells.
Thus, after applying the first mean value theorem for integrals [5] together with well-known properties of the Jacobian elliptic functions [1] to the last two integrals on the right-hand side of Eq. S36, where t * , t * * ∈ [n * T, n * T + T /2] and with sd (·) ≡ sn (·; m) / dn (·; m) being the Jacobian elliptic function of parameter m, one has γT .
x (t * ) /π > δ n * T +T /2 at η = 0 when the Duffing oscillator exhibits chaotic behaviour. It is straightforward to see that F (ϕ, m) presents an absolute maximum (minimum) at m = m impulse max ≃ 0.717, ϕ = π/2 (m = m impulse max ≃ 0.717, ϕ = 3π/2). It is a 2π-periodic function in ϕ, and presents the noteworthy properties (see Fig. S6): In this situation, one lets the elliptic SE act on the Duffing oscillator while holding the remaining parameters constant. For sufficiently small values of η > 0, one expects that both the dissipation work (the integral in Eq. S37) and .
x (t * ) will approximately maintain their initial values (at η = 0) while the function F (ϕ, m) will increase (decrease) from 0 (at ϕ = 0, π), so that, in some cases depending upon the remaining parameters and the sign of .
x (t * * ) x 3 (t * * ), the energy increment just before the chaotic escape existing for η = 0, ∆E, could be sufficiently large and negative to suppress the initial chaotic state in the sense of leading the Duffing oscillator to the basin of a certain periodic attractor. Clearly, the probability of suppressing the initial chaotic state is maximal at m = m impulse max ≃ 0.717, ϕ = π/2 (ϕ = 3π/2) (i.e., when the impulse transmitted by the SE is maximum, cf. Eq. S40), which is in complete agreement with the foregoing MA-based predictions.
Remarkably, we can obtain an useful alternative estimate of the suppressory amplitude, η ′ , by requiring that the sum of the two excitation terms in Eq. S37 be approximately cancelled: x (t * ) .
which presents a single minimum at m = m impulse max ≃ 0.717, while its behaviour, as a function of the shape parameter, is similar to that of the MA-based upper suppressory amplitude (cf. Eq. S23): It is worth noticing that the approximate character of the suppressory condition given by Eq.
S44 prevents us from ensuring that, even in certain cases corresponding to particular values of the initial conditions and system parameters, the SE can effectively lead the Duffing oscillator to some of the two fixed points x = ±β −1/2 , .
x = 0 . Indeed, Eq. S37 tell us that any decrease of the Duffing oscillator's energy over half a period implies a subsequent decrease of the dissipation work over the next half a period, such that this decrease process continues until some of the mismatches of the (approximate) cancellation of the two excitation terms is sufficiently large to compensate the dissipation work in the sense of yielding an increase of the energy, over a certain half a period, and a subsequent energy oscillation later. This means that the steady behaviour becomes a smallamplitude periodic oscillation around some of the fixed points from a certain instant t = n s T , while the corresponding dissipation work is proportional to the action of the periodic orbit in the phase space: where J ≡ 1 2π pdq is the action integral [6]. Alternatively, one can show the same behavior as follows. After linearizing Eq. (2) in the main text around x = ±β −1/2 , one straightforwardly obtains the equation governing the linear stability of the two equilibria: ..
where ω 0 ≡ √ 2 and z ≡ x ∓ β −1/2 , respectively. Equation S48 has the associated energy equation where we introduced the shift t → t+T /4, and hence ϕ → ϕ−π/2, and where E 0 (t) ≡ (1/2) is the energy function of the linearized system. Integration of Eq. S49 over any interval [nT, nT + T /2], n = 0, 1, 2, ..., yields After applying the first mean value theorem for integrals together with well-known properties of the Jacobian elliptic functions to the last two integrals on the right-hand side of Eq. S50, one where t ′ , t ′′ ∈ [nT, nT + T /2]. Note that the suppressory condition given by Eq. S44 implies the approximate cancellation of the sum of the two excitation terms in Eq. S51, and hence the same reasoning applied above to the general energy E can now be directly applied to the small-amplitude energy E 0 (compare Eqs. S37 and S51), thus allowing us to conclude that the regularized smallamplitude periodic oscillations around any of the fixed points x = ±β −1/2 , .

II. NUMERICAL METHODS
In our numerical simulations, we studied the purely deterministic case as well as the robustness of the impulse-induced chaos-control scenario against the presence of additive noise in the Duffing equation: ..
where ξ (t) is a Gaussian white noise with zero mean and ξ On the other hand, we computed period-distribution and isospike diagrams [9]  In isospike diagrams, black is used to represent chaos; i.e., lack of numerically detected periodicity. To represent maxima, we used a palette of 17 colors. Patterns with more than 17 maxima are plotted by recycling the 17 basic colors modulo 17. Period-distribution diagrams are based on computing the period of periodic solutions after a sufficiently long transient (10 4 drive cycles) for each point on a N × N grid with phase difference ϕ and amplitude η along the horizontal and vertical axes. In period-distribution diagrams we used a colour code to detect periodic solutions with periods between T (period-1 solution) and 8T (period-8 solution). In period-distribution diagrams, black is used to represent chaos; i.e., lack of numerically detected periodicity.
We studied the evolution of the regularization regions in the ϕ − η control plane as the impulse transmitted by the SE is changed from its value at m = 0 to its value at an m value very close to 1 by computing LE, isospike, and period-distribution diagrams. For the purely deterministic case, the results are respectively shown in Figs. 4 and 5 of the main text and Fig. S7, while Fig. S8 shows, for the same set of fixed parameters, four illustrative LE diagrams for the Duffing oscillator in the presence of noise (σ > 0). Although the presence of noise gives systematically rise to a decrease, or even a complete elimination, of secondary and minor islands of regularization in the ϕ − η control plane (see Fig. S9), a comparison between the purely deterministic case (σ = 0) and the noisy case (σ > 0) for the same values of the shape parameter (compare Fig. 4 in the main text with Fig. S8) indicates that the impulse-induced chaos-control scenario is robust against the presence of moderate noise.

III. EXPERIMENTAL SETUP
The experimental setup used in our analog implementation of the damped driven Duffing oscillator (Eq. 2 in the main text) is shown in Fig. S10.