Fractional DDRM in Dynamical Systems

• Fractional DDRM is a discrete-time dynamical system that generalizes the classical dissipative standard map by incorporating noninteger derivative orders to introduce intrinsic long-term memory.
• It employs memory kernels based on Riemann–Liouville and Caputo derivatives, allowing the system's current state to depend on the entire weighted history of past states.
• Numerical studies show that even slight deviations from integer order lead to significant changes in bifurcation structures and attractor morphologies, highlighting its relevance in modeling hereditary dynamics.

The fractional dissipative standard map (fractional DDRM) is a discrete-time dynamical system that generalizes the classical dissipative standard map to include derivatives of noninteger order. This leads to systems with intrinsic long-term memory, where the present state evolves depending on the entire weighted history of past states. As the fractional derivative order parameter $\alpha$ departs from integer values, the system exhibits new types of “fractional attractors”—including scroll-like and fractal geometries—that are not present in the classical integer-order regime. Even minute deviations of $\alpha$ from 2 produce marked qualitative changes in the system’s bifurcation structure and attractor morphology (Tarasov et al., 2011).

1. Foundations and Continuous-Time Formulation

The classical dissipative standard map arises from the stroboscopic section of the equation for a kicked, damped rotator: $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$ where $X(t)$ is the angular variable, $q$ is the dissipation, and $\varepsilon$ is the kick amplitude.

Two principal fractional generalizations exist:

• Fractional kick: Introduces a Caputo derivative of noninteger order $\beta$ in the kicked term.
• Fractional inertia: Implements a Riemann–Liouville derivative of order $\alpha$ ($1<\alpha\le2$) in the un-kicked terms with $\beta=\alpha-1$: $\alpha$0 Here, $\alpha$1 and $\alpha$2 denote the Riemann–Liouville and Caputo derivatives, respectively. The fractional DDRM refers to the discrete map derived from the "fractional inertia" variant.

2. Discrete-Time Map Derivation

To construct a time-discrete map, the fractional ODE is reformulated using an auxiliary function $\alpha$3 satisfying $\alpha$4. This recasts the system as: $\alpha$5 Introducing $\alpha$6 and evaluating the effect of each Dirac delta kick yields the discrete-time updates: $\alpha$7 Alternatively, rescaling variables ($\alpha$8) gives the familiar standard–map–like form: $\alpha$9 where $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$0 and $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$1.

3. Memory Kernels and Their Properties

The hallmark of the fractional DDRM is its long-term memory, mathematically encoded via a sum over all previous $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$2 (or $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$3) with weights given by the memory kernel: $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$4 with $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$5 (upper incomplete Gamma function) and $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$6.

As $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$7, the kernel reduces to $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$8, and the map regains the one-step memory of the classic Zaslavsky dissipative standard map. For noninteger $$\ddot X(t)+q\,\dot X(t)=\varepsilon\,\sin X(t)\;\sum_{n=0}^{\infty}\delta(t-n)$$9, the kernel spreads weights over the entire history, resulting in a genuinely nonlocal (in discrete time) system evolution.

4. Dynamical Features and Fractional Attractors

Deviation of the fractional derivative order $X(t)$0 from 2, even minutely, yields qualitatively new dynamical phenomena:

• Ballistic Windows: For integer $X(t)$1 and strong dissipation ($X(t)$2), a broad ballistic window exists near $X(t)$3. Inside this window, attractors exhibit standard dying chaos or period-doubling behavior.
• Morphological Shifts: As $X(t)$4 decreases slightly (e.g., $X(t)$5), the ballistic window narrows and shifts ($X(t)$6–$X(t)$7), closing entirely for $X(t)$8. Attractors morph into multi-scroll configurations with period-8%%%%33$\varepsilon$34$X(t)$35%%%%1 transitions and slow phase drifts.
• Scroll-Type Chaotic Attractors: For moderate reductions in $q$2 ($q$3), one-word, two-scroll, and four-scroll chaotic attractors emerge, which have no analog in the integer-order case.
• Low-Period Collapse: In the limit $q$4, attractors become simple low-period (period-2, period-1) orbits in a manner distinct from both $q$5 fractional maps and the classical integer-order dissipative standard map.

These attractors are characterized by:

• Long-term memory, with the entire history of $q$6 or $q$7 participating,
• Fractal, scroll-like geometry,
• Dynamical windows dependent on both $q$8 and $q$9.

Numerical evidence demonstrates window drift and closure, emergence of new stability islands as $\varepsilon$0, and a rich bifurcation sequence as a function of $\varepsilon$1 and $\varepsilon$2 (Tarasov et al., 2011).

5. Illustrative Parameter Scenarios

Simulations typically fix $\varepsilon$3 ($\varepsilon$4, $\varepsilon$5) and vary both the kick strength $\varepsilon$6 and the fractional order $\varepsilon$7:

$\varepsilon$8	Ballistic Window $\varepsilon$9	Attractor Morphology
$\beta$0	$\beta$1	Classical chaotic/ballistic
$\beta$2	$\beta$3	Multi-scroll, slow-drifting
$\beta$4–$\beta$5	Shrinking/moved	Two-, four-scroll chaotic
$\beta$6	Closes	Period-2, Period-1

For $\beta$7, the dynamics reproduces traditional Zaslavsky map features. For $\beta$8, attractor breakup into complex multi-scroll patterns occurs. Further reductions in $\beta$9 give rise to scroll-type chaos, which transitions to simple periodic attractors for $\alpha$0.

6. Context within Fractional Dynamics

The fractional DDRM demonstrates the significant role memory effects play in low-dimensional dissipative dynamics, extending classic results to fractional calculus. The presence of nonlocal memory kernels rooted in Riemann–Liouville or Caputo derivatives leads to bifurcation cascades and attractor families characterized by their strong dependence on the history of the system, distinguishing them from systems modeled with integer-order derivatives. This framework is applicable to a wide class of physical systems where hereditary, nonlocal, or anomalously diffusive mechanisms govern dissipative transport and forced oscillations (Tarasov et al., 2011).