<p>We study a ring of three Duffing oscillators coupled unidirectionally, focusing on the effects of the coupling strength, time-dependent damping and tracking of the attractors using time series, phase portrait, Fourier transformations, bifurcation diagrams, potential planes, Lyapunov exponents and hysteresis paths for two cases of damping: constant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha =0.4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </math></EquationSource> </InlineEquation> and linearly increasing <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha (t)=\frac{t}{\tau }\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mfrac> <mi>t</mi> <mi>τ</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. For constant damping term the dynamics shows limit cycles, heteroclinic orbits, fixed points, quasiperiodic 2D and 3D torus chaos, and hyperchaos. These behavior shows the existence of hysteresis and an important change in the potential energy. If damping term is dependent of the time <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha = \frac{t}{\tau } \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mfrac> <mi>t</mi> <mi>τ</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, the dynamics of the system shows bottleneck behavior influenced by higher frequencies which persists for a small range of values of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>. In summary, the potential energy in this set of three Duffing oscillators depends strongly of three factors: first the attractor tracking (past memories), second the coupling strength and third the damping term.</p>