Chaos is not an invariant of fractional-order systems: the clock shapes orbits, attractors and bifurcations

Technical Summary: "Chaos is not an invariant of fractional-order systems: the clock shapes orbits, attractors and bifurcations"

Problem Statement
In the qualitative theory of ordinary differential equations (integer-order systems), a fundamental tenet established since Poincaré is that multiplying a vector field by a positive state-dependent function (a "clock") alters only the speed at which orbits are traversed, not the geometry of the orbits themselves. This reparametrization invariance implies that phase portraits, limit cycles, attractors, and bifurcation structures are properties of the vector field alone, independent of the time law.

However, fractional-order systems (specifically those defined via the Caputo derivative) possess a non-local memory kernel that weights the past history of the state. The paper investigates whether the classical reparametrization invariance holds for these systems. Specifically, it asks: if a fractional system $D^\alpha x = f(x)$ is modified to $D^\alpha x = g(x)f(x)$ (where $g(x)$ is a positive state-dependent function), do the solutions describe the same geometric orbits, merely traversed at different rates? The author posits that for fractional orders $\alpha \in (0, 1)$, this invariance fails.

Methodology
The study employs a combination of analytical expansion and numerical verification:

1. Analytical Derivation: The author analyzes the short-time behavior of solutions to the Caputo differential equation $D^\alpha x = F(x)$ using the equivalent Volterra integral equation. They perform a Taylor-like expansion in powers of $t^\alpha$ for both the reference system ($F=f$) and the clocked system ($F=gf$). By comparing the series expansions of the solution $x(t)$ and the reference solution reparametrized by the transformed time $s(t) = \int_0^t g(x(\tau)) d\tau$, they isolate the discrepancy terms.
2. Universal Functions: The analysis yields two universal functions of the order $\alpha$, denoted $c(\alpha)$ and $d(\alpha)$, which quantify the deviation from invariance.
3. Numerical Verification: The theoretical predictions are tested on four distinct systems:
   • A nonlinear oscillator (to verify coefficients $c(\alpha)$ and $d(\alpha)$).
   • The fractional van der Pol oscillator (to examine limit cycle deformation).
   • The fractional Rössler system (to investigate period-doubling cascades and bifurcation thresholds).
   • The fractional Chen system (to test the stability of chaotic regimes and periodic windows).
   • Two numerical schemes were used: the fractional Adams-Bashforth-Moulton predictor-corrector method and a diffusive representation method (superposition of exponential modes) to ensure robustness.

Key Contributions and Results

1. Failure of Clock Covariance: The paper proves that for $0 < \alpha < 1$, the orbits of a fractional system are not invariant under a state-dependent clock.

   • Time Law Discrepancy ($t^{2\alpha}$): At the first non-vanishing order, the difference between the clocked solution and the reparametrized reference solution is tangent to the orbit. This discrepancy is governed by the universal function:
     $$c(\alpha) = \frac{1}{\Gamma(1+2\alpha)} - \frac{\alpha}{\Gamma(1+\alpha)\Gamma(2+\alpha)}$$
     This function vanishes only at $\alpha=1$ (recovering the integer-order theorem) and approaches 1 as $\alpha \to 0$.
   • Geometric Discrepancy ($t^{3\alpha}$): At the next order, a transverse component emerges, proving the orbits are geometrically distinct curves. This is governed by a second universal function $d(\alpha)$, which is non-zero for all $\alpha \in (0, 1)$.
   • Numerical Confirmation: The coefficients $c(\alpha)$ and $d(\alpha)$ were numerically verified with high precision (error < 0.1% for $c$ and < 1% for $d$) against the theoretical predictions.

2. Invariance of Local Stability: The author demonstrates that local properties remain invariant. The equilibria, their linear stability, and Matignon's critical order (the threshold $\alpha^*$ where stability is lost, determined by the argument of eigenvalues) are unchanged by the clock. This is because Matignon's criterion depends on the argument of eigenvalues, which is unaffected by multiplication by a positive scalar $g(x^*)$.

3. Non-Invariance of Global Dynamics: While local stability is preserved, global structures are deformed.

   • Limit Cycles: In the fractional van der Pol oscillator, changing the clock deforms the limit cycle shape and amplitude by up to 16% for intermediate orders, a phenomenon impossible in integer order.
   • Bifurcations: In the fractional Rössler system, the clock shifts the values of local maxima but does not alter the sequence of period-doubling bifurcations or the critical order for chaos onset.
   • Chaos vs. Periodicity: In the fractional Chen system, a state-dependent Arrhenius clock (dependent on temperature-like variable $z$) can transform a periodic regime into a chaotic one (or vice versa) at the same order $\alpha$ and from the same initial condition. Specifically, at $\alpha=0.94$, the system is periodic with a trivial clock but chaotic with a specific Arrhenius clock.

4. General Rule: The paper establishes a rule for the shift of bifurcations:

   • Local bifurcations (equilibria stability, Hopf bifurcations) do not move.
   • Global bifurcations (period doublings, crisis, birth/death of periodic windows) shift linearly with the strength of the clock.
   • The "fragility" of a system to a clock change is determined by its proximity to a global bifurcation.

Significance and Claims
The paper claims that for fractional-order systems, the "clock" is not merely a mathematical convenience but a physical component of the model.

• Physical Interpretation: In integer-order models, multiplying a field by a state-dependent factor (e.g., a temperature-dependent rate) is equivalent to redefining the time unit. In fractional models, this choice is a distinct physical hypothesis:

  • $D^\alpha x = g(x)f(x)$ implies memory accumulates in physical time.
  • $D^\alpha X = f(X)$ in an intrinsic time $s$ implies memory accumulates in intrinsic/material time (a concept used in viscoelasticity and glassy dynamics).
    The paper asserts these are two different models with different predictions, and the difference is quantified by the universal functions $c(\alpha)$ and $d(\alpha)$.

• Implications for Chaos Theory: The author concludes that "chaos is not an invariant" of fractional systems. Minimal orders for chaos, bifurcation diagrams, and the shapes of strange attractors are properties of the specific field-clock pair, not invariants of the system itself. Consequently, the necessary condition for chaos based on Matignon's criterion is robust, but the actual occurrence of chaos depends on the specific formulation of the kinetic factors.

• Practical Consequence: Researchers using fractional models must explicitly declare whether memory accumulates in physical time or an intrinsic material time. A fractional attractor drawn in state space is a property of the vector field and the clock used to run it; a different clock would yield a different attractor, potentially with a different nature (periodic vs. chaotic).