Parameter-Induced Bifurcations in Duffing Systems

The Duffing system is a classic nonlinear dynamical model widely used to study phenomena like chaos and bifurcations. Variations in system parameters lead to distinct dynamical behaviors, particularly the emergence of bifurcation phenomena.

Impact of Excitation Frequency Changes in excitation frequency constitute a primary factor triggering bifurcations in Duffing systems. When the excitation frequency approaches the system's natural frequency, period-doubling bifurcations may occur, eventually leading to chaotic behavior. For instance, at low frequencies, the system may exhibit stable periodic motion, but as frequency increases, it may undergo period-doubling bifurcations into chaotic states. Code implementation typically involves frequency sweep algorithms with ODE solvers (e.g., MATLAB's ode45) to track bifurcation points.

Influence of Linear Stiffness Parameter k1 The linear stiffness parameter k1 determines the restoring force characteristics. When k1 is small, nonlinear terms dominate, potentially yielding stronger nonlinear behaviors like multistability and jump phenomena. Increasing k1 enhances linear effects, stabilizing the system. Near critical values, k1 variations can trigger saddle-node or Hopf bifurcations, transitioning the system from stable equilibrium to oscillations or chaos. Parameter continuation methods (e.g., AUTO or custom Newton-Raphson implementations) are commonly employed to detect these bifurcations numerically.

Computation of Lyapunov Exponents Lyapunov exponents quantify the system's sensitivity to initial conditions—a key indicator of chaotic behavior. For Duffing systems, Lyapunov exponent spectra can be computed numerically using algorithms like the Wolf method. A positive maximum Lyapunov exponent indicates chaos, while zero or negative values correspond to periodic or steady-state motion. Implementation requires simultaneous integration of the system equations and their variational equations, often using orthogonalization techniques (e.g., Gram-Schmidt) to maintain numerical stability.

Effects of Other Parameters Besides excitation frequency and k1, damping coefficients and nonlinear stiffness parameters also influence bifurcation behavior. For example, low damping facilitates chaotic transitions, while variations in nonlinear stiffness may cause symmetry-breaking bifurcations, shifting the system from symmetric periodic solutions to asymmetric ones. Bifurcation diagrams generated through parameter sweeps and Poincaré sections are essential tools for visualizing these transitions programmatically.

By adjusting these parameters and observing bifurcation diagrams or computing Lyapunov exponents, one can gain deep insights into the complex dynamics of Duffing systems, providing crucial foundations for nonlinear system research and control strategies. Numerical approaches often involve phase space reconstruction, time-delay embedding, and recurrence analysis to characterize dynamical regimes.