Controlling birhythmicity in a self-sustained oscillator by time-delayed feedback

Pushpita Ghosh, Shrabani Sen, Syed Shahed Riaz, and Deb Shankar Ray

Phys. Rev. E 83, 036205 – Published 17 March 2011

Abstract

Time-delayed feedback is a practical method for controlling various nonlinear dynamical systems. We consider its influence on the dynamics of a multicycle van der Pol oscillator that is birhythmic in nature. It has been shown that depending on the strength of delay the bifurcation space can be divided into two subspaces for which the dynamical response of the system is generically distinct. We observe an interesting collapse and revival of birhythmicity with the variation of the delay time. Depending on the parameter space the system also exhibits a transition between birhythmicity and monorhythmic behavior. Our analysis of amplitude equation corroborates with the results obtained by numerical simulation of the dynamics.

Received 20 August 2010

DOI:https://doi.org/10.1103/PhysRevE.83.036205

Authors & Affiliations

Pushpita Ghosh¹, Shrabani Sen¹, Syed Shahed Riaz², and Deb Shankar Ray^1,*

¹Indian Association for the Cultivation of Science, Jadavpur, Kolkata, 700 032, India
²Department of Chemistry, Belur Ramakrishna Mission Vidyamandira, Howrah, 711202, India

^*pcdsr@iacs.res.in

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Vol. 83, Iss. 3 — March 2011

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Images

Figure 1
Parameter domain for the existence of a single limit cycle (area denoted by gray region) and the three limit cycles (area denoted by the white region without the dotted line in absence of delayed feedback). The dotted line denotes the bifurcation line showing the two parameter zones corresponding to the points $p_{1} (0.005, 0.144)$ and $p_{2} (0.001, 0.1)$ for two distinct dynamical scenarios in presence of delayed feedback corresponding to Figs. 4 and 5, respectively.Reuse & Permissions
Figure 2
(a) Plot of $f (A^{2})$ versus square of amplitude $(A^{2}$ ) for the values of parameters $μ = 0.1$ , $K = 0.1$ , $α = 0.144$ , and $β = 0.005$ at different time delay ( $τ$ ) ranging from $0$ to $2 π$ . $f (A^{2})$ versus $A^{2}$ curve moves downward for $τ$ values for $0.0 π$ (three roots), $0.3 π$ (one root), $0.6 π$ (no root), or $1.0 π$ (no root) and then moves upward $1.3 π$ (no root), $1.5 π$ (no root), $1.8 π$ (one root), or $2.0 π$ (three roots). (b) Plot of $f (A^{2})$ versus $τ$ for some fixed values of $A^{2}$ with other parameters remaining the same.Reuse & Permissions
Figure 3
(a) Plot of $f (A^{2})$ versus square of amplitude $(A^{2}$ ) for the values of parameters $μ = 0.1$ , $K = 0.1$ , $α = 0.1$ , and $β = 0.001$ at different time delay ( $τ$ ) ranging from $0$ to $2 π$ . (b) The square box of the left-lower corner of Fig. 3a is shown (enlarged) in Fig. 3b. (c) Plot of $f (A^{2})$ versus $τ$ for some fixed values of $A^{2}$ for the parameter set mentioned above.Reuse & Permissions
Figure 4
Control of attractors by delayed feedback. Numerical simulation: Time evolution profile for the parameter values $α = 0.144$ , $β = 0.005$ , $μ = 0.1$ , and $K = 0.1$ for several values of time delay $τ$ with two initial conditions. Dotted line is for the initial values $x_{0} = 1.0, {ẋ}_{0} = 0.0$ , and the solid line for $x_{0} = 8.0, {ẋ}_{0} = 0.0$ . (a) $τ = 0.0 π$ , (b) $τ = 0.2 π$ , (c) $τ = 0.4 π$ , (d) $τ = 1.0 π$ , (e) $τ = 1.6 π$ , (f) $τ = 1.8 π$ , and (g) $τ = 2.0 π$ .Reuse & Permissions
Figure 5
Control of attractors by delayed feedback: Numerical simulation: Time evolution profile for the parameter values $α = 0.1$ , $β = 0.001$ , $μ = 0.1$ , and $K = 0.1$ for several values of time delay $τ$ with two initial conditions: dotted line is for the initial values $x_{0} = 1.0, {ẋ}_{0} = 0.0$ , and the solid line for $x_{0} = 8.0, {ẋ}_{0} = 0.0$ . (a) $τ = 0.0 π$ , (b) $τ = 0.2 π$ , (c) $τ = 0.4 π$ , (d) $τ = 0.598 π$ , (e) $τ = 1.6 π$ , and (f) $τ = 1.8 π$ .Reuse & Permissions
Figure 6
Control of attractors by delayed feedback. Numerical simulation: Time evolution profile for the parameter values $α = 0.145$ , $β = 0.005$ , $μ = 0.1$ , and $K = 0.12$ for several values of time delay $τ$ with two initial conditions: dotted line is for the initial values $x_{0} = 1.0, {ẋ}_{0} = 0.0$ , and the solid line for $x_{0} = 10.0, {ẋ}_{0} = 0.0$ . (a) $τ = 0.0 π$ , (b) $τ = 0.16 π$ , (c) $τ = 0.5 π$ , (d) $τ = 1.48 π$ , (e) $τ = 1.8 π$ , and (f) $τ = 2.0 π$ .Reuse & Permissions

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